I always assumed that the “covering” God provides Adam and Eve in Genesis 3 is a picture of atonement. Like the pitch that covered the ark.
However, that’s not the word used here. A "covering of skin" is a phrase that shows up again later, in Genesis 27. The usage of the phrase may give us a clue.
Then Rebekah took the best clothes of Esau her older son, which she had in the house, and put them on her younger son Jacob. She also covered his hands and the smooth part of his neck with the goatskins.
Genesis 27:15-16 (NIV)
Perhaps God is doing something… tricky.