Remark

Suppose that the field D is commutative. By definition, B(V) is a subset of $V^n$. It is clear that the restriction to B(V) of a nonzero alternating $n$-linear form $f: V^n\rightarrow D$ belongs to Ω(V). Moreover, if $(e_1, . . . , e_n)$ is a basis of V and $t$ is a nonzero element of D, then there exists a unique alternating $n$-linear form $f$ such that $f(e_1, . . . , e_n) =t$ (III, §7, No. 4, p. 511 and III, §7, No. 8, p. 518). By Corollary 1, the set Ω(V) consists of the restrictions to B(V) of the nonzero alternating $n$-linear forms.