If you have a spare skirt hanger, you can hang your mat in the closet. A piece of thick plywood or wood in general could work as a makeshift cutting surface. Take your time with cutting and accuracy will improve. My problem is not the cutting but maintaining the 1/4 inch seam. I am getting my husband's desk tomorrow when he replaces his desktop computer with a laptop... woohoo.
Don't throw out your "broken" cutting mat. Cut it into a smaller useable surface and use it to cut out stuff with templates like triangles or tumbler blocks. Some people use them to line the bottom of quilted bags that they make.