An egg can bind, add moisture, emulsify, or create structure through trapped air. A spoonful of ground flax mixed with water does not perform all those jobs in the same way.
A common starting mixture is one tablespoon ground flaxseed with three tablespoons water, left to thicken. It can be useful in some muffins, cookies, and other recipes designed around that behavior. Follow a tested recipe's instructions rather than applying the ratio indiscriminately to every cake.
Whole flaxseed does not thicken in the same way as a fine meal. Old ground seed can also bring an unpleasant flavor, so store it as directed and check its condition.
For a recipe that relies on whipped egg whites or several eggs for its structure, begin with a deliberately vegan version. You will learn more from a workable formulation than from replacing each egg in a complicated bake and hoping the rest remains unchanged.
Further reading: Bob’s Red Mill: flaxseed meal egg replacer.