Was not this appropriation of the skin of the burnt offering to the priest, typical of the clothing of our souls with the garment of JESUS? If so, it is worthy remark that in the very first sacrifice in the garden of Eden this was taught. See Genesis 3:21. And again Genesis 27:16.
Related Commentaries of Leviticus 7:6
Leviticus 7:6-8
6 Every male among the priests shall eat thereof: it shall be eaten in the holy place: it is most holy.
7 As the sin offering is, so is the trespass offering: there is one law for them: the priest that maketh atonement therewith shall have it.
8 And the priest that offereth any man's burnt offering, even the priest shall have to himself the skin of the burnt offering which he hath offered.