Skins game (golf). Origin?

[Sam Clements]

Other than a fanciful story about ancients golfers in Scotland betting real
skins on the outcome of games, can anyone give a cite.

The OED doesn't list, nor Mathews.  The OED does list "skin game" in the
1860's meaning a gambling game which is rigged towards the house.  And DARE
list playing "skin" meaning a card game in the 1920's.  But these meaning
may have nothing to do with the golf term.

Could it be as simple as a bunch of Americans in the 20th Century betting a
dollar per hole, and the "skins" referring to the dollars that were bet?

Sam Clements