I would consider just building a 2x4 or 2x6 joist floor there. This would allow you to put insulation in the floor so it will be much more comfortable.
Remember, there are very important code considerations regarding the intersection of garage and living space. You will need to pour a concrete curb under the new wall that will separate the garage from your new room. You will also need to make sure that you use the proper fire code drywall and common door. You will also need to make sure that there is no risk of carbon monoxide or other fumes getting into the house through ducts, etc. I wouldn't do it without a permit and inspection, usually required for this type of job. Also, realize that 20' is minimal length for the inside of a garage. It is right at the point where a full size van or P/U might not fit. Not a huge deal, just fyi.