My frame was not hidden from you,We can't skim over a verse like this; not the way things are in the world today.
when I was being made in secret,
intricately woven in the depths of the earth.
David is here stating that there was a time that he was in the womb. He talks about when he was being made in secret. That secret place that he is referring to is in the womb. With current technology, we can get a glimpse into that womb, but they couldn't. We can see now that those babies are babies.
The question is: Do you suddenly become you when you are born? Or you could state it: Does the soul enter the body when the baby pops out? The answer to this question is "No" and here is a scripture which, in a small way, validates that thought.
Keep in mind also that if you were being "intricately woven together" that implies a weaver. We know that it is God himself that was working on you.