Well, actually we probably could. And I find this a puzzle myself, why research into this isn't given more priority. First, if it is just oxygen you don't need plants, a strain of green algae called Chlorella which apparently is ideal for this, takes up CO2, produces oxygen, has very little by way of any other requirements except sunlight. Then in a later development, they now use Spirulina which works like Chlorella but is also edible by humans, which Cholorella is not.
There were early Russian experiments that showed that you just need eight square meters of algae to provide all the oxygen needed for a human being.
That may seem a lot, but algae can be stacked in trays above each other and then you use artificial light for the illumination. This worked just fine on the ground, supplied Russian volunteers with enough oxygen to breath throughout the six months duration of their longest experiments. They have also been able to grow some of their own food as well in these simulated closed system conditions.
And - modern LED lights like this are about an order of magnitude more efficient than the earlier ones
I figured out, a modern LED grow light like this would need about 4.8 kW to supply lighting for the algae for a crew of six.
This is the early Russian experiment which used Chlorella - which is great for producing algae but is more or less inedible.
More recent experiments use Spirulina
This is the idea of the Melissa project. It differs from BIOS-3, in more use of microbes and - basically high grade composting. While for BIOS-3 they burn the plant wastes in an oven. Both approaches have advantages and disadvantages.
So the ISS would seem to have enough power for lighting to supply to the algae - which you would do with light pipes or similar.
The actual volume of algae needed for one person is very small.
You could also use light collectors like this to collect light from outside the ISS and pipe it into your algae tank using light tubes.
That then would use no power at all, when the ISS is in daylight, as it is a lot of the time.
This would work especially well for BIOS-3, not so useful for Melissa.
This is something that the ESA is testing in space.
Algae and plants can behave differently in orbit. And there is a lot more to life support than just producing oxygen and removing CO2. But it's an important part and I don't understand myself why this Russian system, proved to work on the ground, has never been tested in space.
There was a news story just today about the astronauts eating the first ever food grown in orbit and eaten there (until now they have grown plants but not been permitted to eat them as it was for research). But this is such a long way away from what has been achieved in these ground based simulations, with all the oxygen and a fair amount of food all produced from plants.
They could grow all their food also, if we find suitable crops for zero g - or else find a way to supply low g levels (say by spinning the containers with the crops).
This shows the Russian experiments again - in the BIOS3 closed system experiments, they produced nearly all their food - needing some supplement, but most of the bulky stuff - from just 13 square meters per person.
So - it's not so hard to produce food also in space. Using hydroponics or aeroponics - high tech methods - but they are used on the ground also, for agriculture, so not so hugely expensive, and in space the expense would be a small part of the cost of getting it up there.
Also - plants tend to be perceived as unreliable - but - we aren't talking about your allotment where a pest or disease of plants can decimate a crop if unlucky. In the ISS or another such system, there are no pests or diseases of plants. Even on Earth in hydroponic facilities, the facilities can be sterilized of them.
So then - plants are in many ways more reliable than machines. Even if your entire algae crop dies through some malfunction - so long as you have a few viable cells, you can grow it again. Try doing that if your machines for making oxygen break down.
On the assumption that the ISS uses solar power for BIOS-3 type system, the power requirements, she found, were
ISS: 1.2 kW MELiSSA 4 kW BIOS-3 0.92 kW
She finds that the ISS system has least startup mass, of 1.773 metric tons, but much higher supply mass per day of 6.5 kg.
Melissa has most startup mass of 15.711 tons, and supply per day 1.4 kg
BIOS-3 has 6.250 startup mass, and 0.5 kg supply per day.
BIOS-3 if it can be got to work on the ISS, breaks even after two years.
MELISSA breaks even after 7.5 years. But with more use of short lived bacteria, it is more easily controllable.
In any case BIOS-3 was designed for planetary surface habitats originally and not tested in space. So it would need some work for zero g, testing to find which of the plants they used will grow in zero g and what the performance is like - or else you use centrifuges for gravity in space.
MELiSSA is in the early stages of testing in the ISS with small scale experiments with the algae.
(have just updated this answer, 27th Sept 2015 - complete rewrite, after the research into the literature I did for the Science20 article and kindle ebook)