Alternating k-Linear Form on Euclidean Space

Let $n,k\in\mathbb{N}$. A function

$$\omega:(\mathbb{R}^n)^k\to\mathbb{R}$$

is called a $k$-linear form on $\mathbb{R}^n$ if for each index $r\in\{1,\dots,k\}$, for every choice of vectors $v_1,\dots,v_{r-1},u,w,v_{r+1},\dots,v_k\in\mathbb{R}^n$, and for every scalars $\alpha,\beta\in\mathbb{R}$, one has

$$\omega(v_1,\dots,v_{r-1},\alpha u+\beta w,v_{r+1},\dots,v_k) =\alpha\,\omega(v_1,\dots,v_{r-1},u,v_{r+1},\dots,v_k) +\beta\,\omega(v_1,\dots,v_{r-1},w,v_{r+1},\dots,v_k).$$

A $k$-linear form $\omega$ is called alternating if

$$\omega(v_1,\dots,v_k)=0$$

whenever $v_p=v_q$ for some distinct indices $p,q\in\{1,\dots,k\}$. An alternating $k$-linear form on $\mathbb{R}^n$ is also called an alternating covariant $k$-tensor on $\mathbb{R}^n$.