We have one already. At least probably. That's Chroococcidiopsis.
Some strains have withstood 528 days exposure to Mars simulation conditions on the exterior of the ISS.
And the DLR (German space aviation company) is undergoing a series of experiments with lichens and cyanobacteria in Mars simulation experiments. They found that not only do they survive - they can actually photosynthesize and metabolize in the Mars atmosphere using just the night time humidity.
As you can see there, lichens may be able to survive on Mars also.
So far these are relatively short duration experiments.
There is also good evidence of liquid water on Mars at least sporadically in geologically recent past from the Phoenix isotope results. And indirect evidence suggesting there probably are micro habitats of liquid water there right now - thin layers and droplets a few mm across, for instance on interface between ice and salt, that form for a few hours a year.
This means of course, that we can be more optimistic also about the possibility of finding life on Mars.
I think we should be cautious about introducing life to the surface of Mars for the time being however, and indeed it is Planetary Protection policy at the moment to keep Mars free of Earth life so that we can study it as it is now.
Terraforming is not as easy and straightforward as you might think from some science fiction stories. It took millions of years on the Earth, and for Mars, the most optimistic suggest a thousand years to get to the point where you have extensive Earth like vegetation on Mars, without oxygen, and a few more millennia to get an oxygen rich atmosphere (that's the Mars Society, who have as their objective Mars surface colonization).
Also Mars is different from Earth also in many ways.
For some of the things that could go wrong and some of the issues:
And before you can make significant amounts of oxygen from the Mars atmosphere, then you need to capture the carbon, not just cycle it through, because photosynthesis works by carbon capture, to a thickness of several meters over entire surface of Mars. If you do it just by covering Mars with green vegetation that's likely to take many millennia possibly 100,000 years - and that's if it worked, with many things that could go wrong and need for high technology for those thousands of years (e.g. space mirrors or greenhouse gas generating factories or probably both to keep Mars warm enough).
So that's not as easy as imagined in some of the science fiction stories.
MAKING ALL THE OXYGEN AND FOOD YOU NEED IN A HABITAT
As for using them to create oxygen inside of habitats, theoretically 8 square meters of algae is enough to produce enough oxygen for one human being, and 40 square meters is enough to produce all the food and oxygen for one human.
However such ideas have never yet been tested in space.
If we started on that now, then maybe it wouldn't be that long before we also crack the remaining problems and have Biosphere II type installations in space more or less completely self contained.
However so far we haven't yet flown any experiments to space that generate all the oxygen a human needs from algae or with the astronauts eating food grown in space. Some experiments with plants grown in the ISS but no actual tests involving humans breathing oxygen generated by algae and eating food grown in space, AFAIK - certainly no attempt yet at a closed or semi closed habitat of this type in space.