We will search for the answer all our life but the question we ask at each stage of our life would be different. As we grow older, a new question will emerge and overlay the earlier question regarding who we are.
You will notice that the questions we ask shift from being self-centred to being other-centred as we grow older. As Gandhi said "The best way to find yourself is to lose yourself in the service of others."
We don't have to wait till we are old before we ask the other-centred questions regarding Who Am I? We don't have to wait for the questions to occur to us one by one as we arrive at each stage of life. Some people may even get stuck at the self-centred stage although they may grow older in age - they never get around to asking the higher order, other-centred questions.
We can ask all the questions all at once. We will be wiser sooner if we ask the other-centred questions earlier in our life.