Similar topic but different concern with this
In the Exterior_algebra of Wikipedia, there is:
Under this identification, the exterior product takes a concrete form: it produces a new anti-symmetric map from two given ones. Suppose $ω$ : $V_k \to K$ and $η$ : $V_m \to K$ are two anti-symmetric forms. As in the case of tensor products of multilinear maps, the number of variables of their exterior product is the sum of the numbers of their variables. It is defined as follows: $$ω\wedge η=\frac{(k+m)!}{k!m!}Alt(ω\otimes η) $$
Here are confusions about the definition:
(1) Both $ω$ and $η$ are alternating multilinear forms, their tensor product $ω\otimes η$ should be a multilinear map whose image space is $K\otimes K$ instead of $K$, then the result of the above formular should also be such multilinear map which is not multilinear form anymore (guess not)?
If no, then I guess $ω$ and $η$ in the above formular are any representative tensor in $T^k(V^*)$ for their corresponding elements in $\Lambda^k(V^*)$? Since this is "compatible" with the common definition of the map $Alt$ which map tensor to tensor.
(2) How is the coefficient $\frac{(k+m)!}{k!m!}$ produced? The note in the article says this is only a convention. Also note that the $Alt$ in the article already includes the $\frac{1}{k!}$ factor, then there is double copy of the $\frac{1}{k!}$ factor. Also there is no such factor in $Alt$ in Dummit's book.