I swam with eczema throughout the time I was competing (and it was the reason I quit for rowing when I went to uni, was just fed up of the itching due to the really high level chlorine pools around me). What I found out to work best was to apply a moisturiser (I used Diprobase), about an hour before I went swimming, everywhere except around the eyes. Once I had finished I immediately showered, using a soap substitute, patted rather than rubbed dry and put another layer of cream on. The faster I showered, dried and applied that second layer, the better the skin felt.

Worth noting that it might well be worth having a sacrificial long sleeve top for after you have swum if doing stuff like that as the moisturiser soaks in. Especially important if using creams like diprobase as it is parafin based and if you for some reason stand near an open flame with clothing that has absorbed some you risk becoming the Human Torch.