Elbows and planks

I'm wondering if anyone else has a similar issue: the skin on my elbows is looking more and more red and almost grazed, and it gets worse after workouts that include planks or any other exercises that rely on elbow / forearm support.

I have a pretty thick yoga mat, but it doesn't seem to help much. I mean, without it I wouldn't be able to do any floor-based exercises because I have hard marble tiles in this apartment, but it doesn't seem to be enough for my elbows. Any idea for what to do?

My plank holding times are extending (yay!) so the problem will be getting worse and worse ;-)