I'm not an expert, but it sound to me like it might be contact dermatitis, something I also had on my hand/wrist. It started out as eczema (an allergic reaction to my rings) and just got worse. My doctor gave me prescription hydrocortisone gel, which I applied every night for 2 weeks. You then need to take 2 weeks off (since hydrocortisone is a steroid cream) and then you can do it again for 2 weeks.
She also said that I should make sure that my hands were completely dry after washing them, and to use a hand cream (like Neutrogena Norwegian formula hand cream --- Eucerin and Aquafor make similar products) after every time I washed my hands.
It was always worse in winter.
You could probably try doing this with over-the-counter hydrocortisone, but I would recommend seeing a doctor if it doesn't get better in a couple of weeks.