1 1/2 TBL water
1 1/2 TBL oil
1 tsp baking soda
1/2 tsp vinegar (optional)

Whisk above ingredients together in a cup and pour into mixture that calls for an egg.

Note: this only works as a substitute in baking items like cakes, muffins, etc.- things that need to rise.
It gives the chemical components needed for rising, but no flavor whatsoever. It would not work in a cheesecake because you need the egg yolk/flavor for that.
Adding the vinegar should provide a bit more rising power.