I STILL tear off skin on a regular basis by running into walls, chairs, everything. I am pretty clumsy and my fragile skin just makes it look worse than it is.
First, my best friend is COHESIVE bandage, which only sticks to itself. It basically like ace bandaging but it helps keep a gauze pad over a wound and keeps the bactracin in place.
Second, my Mom learned early on to pack a first aid kit for me and leave it at the school office every school year. They were constantly sticking band-aids on me even when I told them not to. Frankly, my Mom taught me this white lie: I am allergic to adhesive and cannot have it touching me. Schools are so freaked out about kids and allergies that they stopped sticking stuff to me. It's kind of a last ditch thing to say when they are coming at you with a band-aid and it's not a total lie, adhesive gave me horrible infections, worse than the boo-boo the band-aid was covering.
I think that packing your kids first aid kit is one of the best things that you can do for them, by the time they start going to school they will know how to treat their injuries better than anyone else.