First, why is it scratching? Are the feet missing? Is the battery bulging? (If it is, you need to stop using it.) The bottom of the Mac should be flat, and only touching on the feet, with an air space between the bottom and the desk. So the first thing to do is to determine what is wrong.
A cloth mat is not a good idea, as it could interfere with the Mac's cooling itself through the aluminum skin. Something hard and flat would be a better choice, such as a large blotter pad or cutting board or something like that. Maybe just a piece of poster board. Alternatively, you could place a couple of flat erasers by the rear feet and elevate the rear by 1/2" or so, allowing for good air flow underneath.
The Mac does radiate heat through the aluminum skin as a means of cooling itself, so you do need to have it on something that will allow the air to flow freely.
Good luck!