Will the wound on my toe heal on its own?

My daughter did the same thing a week ago. Just clean it and then let it air dry. The skin will get kinda rough from lack of moisture then take a pair of nail clippers and clip it off. Then keep it clean, neosporin band-aid and then take it off at night you'll be okay in a few days, if it gets really sore though get it checked but if you do what I said you should be good.