Dishwashing detergent (and shampoo, and soap, etc…) all have a detergent called Sodium Laurel Sulfate in it that will dry out your skin. It strips the oils and disrupts the proteins in your skin cells.
Also, the dry winter air means that the skin cells lose a lot more moisture than they normally would. This will speed up the programmed demise of these cells because the loss of water decreases or hinders the cells’ normal functions.
So try and avoid using too many harsh detergents (and things with Sodium Laurel Sulfate) or at least wear gloves while doing dishes. Also, make sure you put on a lot of lotion that has some sort of “grease” to it: shea butter, cocoa butter, petroleum, etc… this will help to create a water barrier to keep the moisture in and also will be the first to go if you come in contact with detergents (instead of your skin’s natural proteins and oils).
When things get really bad, slather “greasy” lotion on at night and sleep with cotton gloves on to make sure i hydrate the top layer of skin as much as possible.
Sorry for sounding like a commercial, but i hope this helps!