What is the formal definition of a Cayley table?

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What is the formal definition of a Cayley table? I am not interested merely in Cayley tables for groups, I am interested in general Cayley tables for non-empty finite magmas. Also, another question is, does the ordering of the elements in the row and column have to be the same? Like, for example, if $a,b,c$ are the $3$ elements of a magma, can we have a Cayley table where the order of the row is $(a,b,c)$, while the order of the column is $(b,a,c)$? I would like it if there was a formalization of Cayley tables for non-empty finite magmas.