I'm very sorry that you are having so much trouble. I do think it may at least partly be caused the drier air in winter. Dry air depletes the moisture in the skin and can aggravate eczema. It helps to use a lot of moisturizers. Neutrogena Aquafor is very good, and I have a new one from Walgreens called Moisture Recovery Lotion which is great, and cheap, too ($.99).
Who knows what causes eczema. I've read some theories that it may be caused by a skin fungus. If this is so, Antibiotics and bleach baths won't help. I'm not a doctor, though, so I have no basis to say this other than my belief that if this is a possibility, then I want to exclude any remedy that would aggravate a fungus.
I used to have eczema on my hands. It started several years ago when I began using tea tree oil soap, which I must have been allergic to. It got very bad because it took me a long time to pin it down to the tea tree oil soap. Even after I stopped using it, though, it took years for the eczema bouts to end.
What I would suggest is that you begin reading as many posts here as possible to learn what has helped others. Begin on this page and keep going back and trying different suggestions or insights until you find what helps for you. Some people have found cures with very simple solutions such as eating a lot of watermelon. Goodsamaratin here found relief from eating a raw diet. My cure was Doctors Defense Zen Fusion Creme. These posts are a treasure trove of information because they document through the years what others have found helpful.