All the karate dojos I've ever trained in have been! We had to clean them the traditional way after each lesson - all line up at one end with a rag in our hands and crawl our way down the dojo pushing the dust to the end with the cloth, then scoop it up and tip it out the window.
I'm not sure on tae kwon do, as I've always done karate, but presumably you do a lot of twisting and pivoting on your feet? Perhaps it's the way in which you're moving on them, could you try and use a slightly different stance or put your weight down slightly differently and see if that helps?
As a short term measure, I'd try taping my toes on the problem areas so they slide more easily and the skin remains intact until it heals. Though obviously don't tape the entire thing or wear socks as that can get very dangerous!
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