The most common medical grade skin glue is Dermabond, and like most medical supplies, it is extremely expensive. Typically, it is for sale only through a medical wholesaler, but a quick google search yields online retailers willing to sell it to you.
I'm not a big fan of putting toxic chems on my cuts. I did get in touch with a friend who is getting me some through her vet clinic, there is potential for bulk orders if anyone is interested, not sure the price.
Well these glues typically work best for cracks and lacerations with intact edges so you shouldn't be putting any of them in your wound. The idea is to hold the skin edges together. I have a fair bit of experience using both medical grade and common pharmacy skin glues, and have found all of them to be underwhelming. OK for hang nails and cuticles, but otherwise pretty useless - especially for abrasions and punctures typical of climbing. They are alright if combined with lots of tape and you are trying to salvage a session, but the wound coverage is just not very durable.