I have the exact problem you are describing. I will tell you what I do to help the problem. First, let me state the first rule of Aikido and bandages. There isn't a band-aid or tape made that will stay put on your toes or fingertips while on the mat! Most times it'll fly right off during warm-ups.
The only thing that I've found that will stick to toes or fingertips is a product called NuSkin. If you can find it I highly recommend it. It's the only thing that works for me.
You'll want to do a little preparation first. I use a q-tip dipped in rubbing alcohol to clean the toe. This may sting a bit. Then I will take ordinary nail clippers and either dip them in the alcohol or swab them down with it to sterilize them. I clip the skin that's hanging off in a small flap. It's easier after showering while the skin is still soft. Be careful to only cut the dead skin off! I find that this helps in preventing the flap catching on the mat, or anything else, and traveling further up the toe.
When the toe is completely dry. I will then put two coats of the NuSkin on the area around were the flap was. This may also sting a bit at first as well. My toe is good for the day's practice after that.
Please keep in mind that I am not a doctor and I am only relating what works for me. If you have any pain or bleeding then I would do as others have suggested and go to a qualified health professional.