There aren't many multilinear algebra textbooks,even older ones. The wonderful text on algebra by E.B. Vinberg has a terrific chapter on it. Volume 1 of the treatise by Anthony Knapp has a very good chapter on it, more complete then Vinberg's.
As for actual whole textbooks, there are basically 3 of them: For multilinear algebra from a purely algebraic and formal point of view, there's the classic textbook by W.Grueb, Multilinear Algebra. which is very austere but comprehensive. Less difficult is the book of the same title by Northcott (a nearly forgotten algebraicist which I'd love to help republish the textbooks of one day). Lastly, there's a good discussion in the advanced linear algebra text of T.Y.Blyth, Module Theory.
As far as I know,that's all there is as far as "standard sources" go.