So there's no need for me to actually exchange the infamous coat for linen in order to actually have those commodities related to each other as values, I presume?
There is no need; in free access socialism 5 yards of Linen will contain the same amount of human effort (abstract labour), albeit of different ‘concrete’ types, as 1 coat.
The value of an artificial product, denominated in labour, is not dependent on exchange, it is a defined and predicated/premised intrinsic property of the material.
The general value form is the reduction of all kinds of actual labour to their common character of being human labour generally, of being the expenditure of human labour power.
The general value form, which represents all products of labour as mere congelations of undifferentiated human labour, shows by its very structure that it is the social resumé of the world of commodities. That form consequently makes it indisputably evident that in the world of commodities the character possessed by all labour of being human labour constitutes its specific social character.
Source: Also Capital Vol 1 Ch 1
Maybe Stalin was right and the law of value DOES operate under socialism. I thought to do away with it was the very point though?