Dry skin on the hand during the winter is a very common problem, especially in those who have a genetic tendency to have sensitive skin. The cold air and lack of humidity have an effect on the skin's natural layer of protective oils, which leads to drying out, itching, and cracking.
The first step in treating this problem is to eliminate anything you are doing that further strips off the skin's protective oils. Exposure to water is number one here. You should wear plastic gloves whenever washing dishes. You should also avoid washing your hands excessively, and you should always pat them dry, never rub them. You should also limit bathing and showering, typically to no more than once a day and never with very hot water.
Follow this up with very frequent (6 or more times per day) application of a skin moisturizer. The best moisturizers are 'ointments' - those that have a texture like vaseline - because they stay on longer and are better at trapping in the skin's moisture. If you have any areas of eczema
(red, itchy, flaky areas) then you can also apply an over the counter steroid cream to these areas.
If none of these strategies work, then you should seek the advice of your dermatologist