I keep mine on the left side of my sewing table, usually buried under "stuff". OR behind the sewing table, leaning up against the wall. I usually keep something sitting in front of it to keep it from bending over and warping.
My biggest one doesn't have holes for hanging. My bigger one is only like 18 x 24.
I read a very good tip on how to store these once, and I think it was a thread on this board but not sure. Anyway, the lady said she used a plastic clothes hanger, like the kind with the clips pants usually come on, and hangs the hanger up with the cutting mat clipped in it! If you don't have one of those, you could always use a regular hanger and clothes pins.