For eczema I recommend calendula oil or jojoba, as well as a few drops of german cham and lavender Also, rosehip oil has a good effect - DS has had some patches lately and when I first didn't know what it was it went red and inflamed, and after making up this mixture, within three days it had gone away. Calendula oil was about $12, Jojoba $19 and about $12 for each essential oil...and it would make about 8-10 25 ml dropper bottles! Might seem like a lot, but it works out in the end.
In the early days, we just used extra virgin olive oil, or sweet almond oil in the bath.