Exercise 9, § 7

Let E be an A-module. The skewsymmetrization of a linear mapping g of $T^n(E)$ into an A-module F is by definition the linear mapping $ag$ of $T^n(E)$ into F such that $ag(z) = g(az)$ for all $z \in T^n(E)$; if $\phi : E^n \to T^n(E)$ is the canonical mapping, the n-linear mapping $ag \circ \phi$ is also called the skewsymmetrization of the n-linear mapping $g \circ \phi$. Deduce from Exercise 8(a) that if E is free every alternating n-linear mapping of $E^n$ into F is the skewsymmetrization of an n-linear mapping of $E^n$ into F.