My hubby, son, and I have eczema. My son's eczema is the worst.
I keep him in long sleeves and pants most of the time to keep contact to whatever allergens to the minimum. Whenever he wears shorts, his legs rashes return.
I don't use soap or cleaning bars on him every time he baths. Baths are short to not dry out his skin.
There is a baby balm that we use at a minimum to help when he does have a rash breakout. It's Mama Rose's Naturals
Magic Touch Baby Skin Balm.
For my skin, I like California Baby
Calming Botanical Moisturizing Cream.
I don't believe in using moisturizers/creams/etc. everyday. Skin absorbs everything, so if the moisturizer's ingredients aren't edible then they shouldn't really be applied to the skin (in my opinion).
Also, some lotions will say to apply after bath on wet skin. That does not work for me. After the bath, I make sure my skin is completely dry before using anything.
You may consider getting a water filter for your shower/bath. I believe it's the water impurities (fluorides, lead, chlorine, etc.) that cause eczema. Don't believe me? Try not taking a bath for a few days and see if you have eczema relief (of course, you'll be stinky).