Yes I know first hand how much eczema on the face sucks.
So here is something I did and I finally have a face that has no acne or eczema and a much happier scalp.
First of: Never use anything drying on the face. I use Pears Glycerin soap or Charcoal & Clay soap from Etsy.com and a cool washcloth.
Second: Always moisturize DIRECTLY after rinsing the suds of your face. I use Shea Butter, Almond oil, or Aveeno Naturals.
Third: Always use a gentle 2.5% Benzoyl Peroxide lotion on existing and in acne prone places. ONLY once a day!!! I cannot stress this enough. Too many applications can exacerbate acne AND eczema.
Final: Drink lots of water. and Watch what you use on your HEAD. Shampoo, Hair color, Sprays,gels,mousses, can all cause your scalp and face to flare up. I have found Chagrin Valley Shampoo bars or Hesh Ayumix Shikakai Shampoo bars to be the best for me.
Good luck and don't scratch it!