Let $V$ be a real vector space and let $k\in\mathbb N$. An alternating $k$-linear form on $V$ is a map
\begin{align}
\alpha: V^k \to \mathbb R
\end{align}
that is linear in each argument and satisfies
\begin{align}
\alpha(v_1,\dots,v_i,\dots,v_j,\dots,v_k)
= -\alpha(v_1,\dots,v_j,\dots,v_i,\dots,v_k)
\end{align}
for all $v_1,\dots,v_k\in V$ and all $i\neq j$.