I have yet to find any satisfactory 'tradition' which adequately explains the absence of designation of the actual animal used by God in this instance. There is a lot of conjecture expressed about the matter but none that I have yet found to be accurate, only superstitious.
The traditions which I have found all attempt to improvise on what is not actually expressed by the scripture - and therefore miss the whole point of the allegory.
There is no information given in scripture as to the species which God used in order to clothe Adam and Eve with coats (kethoneth, long sleeved tunic, Strong 3801 ) of skins. It was an undefined species. The lack of information is highlighted by not being there.
Unto Adam also and to his wife did the LORD God make coats of skins, and clothed them. [Genesis 3:21, KJV.]
My own understanding of the lack of information is that what was to clothe humanity in the future did not - yet - exist. Therefore it could not - yet - be expressed. Therefore it is left un-defined in allegory.
That which clothes humanity unto righteousness is the humanity in which Christ came, as Head of a new humanity.
'Clothing' - the 'robe of righteousness' mentioned by the prophet Isaiah (61:10), and mentioned by the Apostle John in Revelation, the 'righteousnesses of the saints' being 'fine linen' (19:8) - is a matter of clothing the human soul with a new humanity, in righteousness.
For as in Adam all die, even so in Christ shall all be made alive. [I Corinthians 15:22, KJV.]
In order to be so clothed, in the 'skin' of Another, that 'other' must die, in sacrifice.