I have developed a skin irritation on my hand that has spread. it sometimes itches and oozes clear discharge. it gets really dry and splits.

Eczema or dermatitis: You should upload a picture of the rash. This sounds like it might be eczema or eczematous dermatitis. Care hand washing with baby soap. Try over the counter 1% Hydrocortisone cram topically am/pm for 1 week; however, if not resolving, you should be seen by a physician. Do not keep putting on antibiotic ointment until the liquid has been evaluated by culture.