Please refer this chart for Egg substitutes for Cakes (brownies, cookies and rest)
Personally I would go with Almond milk or Coconut milk. But you can use equal amount of Soy milk or rice milk. Depending on the recipe, equal amounts of water or juice would work as well.
You might find these related links useful: 1. Baking Substitutes 101 Chart
2. Tips to Successful Egg Substitution
3. How to find out the role/purpose of eggs in a recipe?
By Rizzah on Nov 15, 2014