When buying a detergent, look for something that is not scented. If you are having trouble finding one that works for you, look for products labeled "free and clear" and see if they work. In addition, it may be best not to use a fabric softener at all. This product can cause an issue for people with eczema, so it is usually not worth the risk.
The cold weather can damage your skin, particularly the skin on your hands. When you are outside in the winter, make sure you put on a pair of gloves. In particular, either cotton gloves or leather gloves will work the best. Stay away from wool gloves, as they may hurt more than they will help you.