Do not scratch your eczema. It is hard to resist because eczema is an itchy skin condition, but scratching makes it worse. You can increase the itching, damage your skin, and increase your risk of getting an infection. Try keeping the skin moisturized, bathing regularly, and using cold compresses to alleviate itching.
Don't cut off all exposure to the sun. Your eczema could be a result of a Vitamin D deficiency, so blocking out the sun entirely is not a good idea. You should be getting at least 10 to 15 minutes of sunlight a day sans sunscreen. Don't get any more than that though or you could burn.