Differential k-Form on an Open Subset of Euclidean Space

Let $n\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be open, and let $k\in\mathbb{N}\cup\{0\}$. A differential $k$-form on $U$ is an assignment $\omega$ which to each point $x\in U$ assigns an alternating $k$-linear form $\omega_x$ on $\mathbb{R}^n$. For vectors $v_1,\dots,v_k\in\mathbb{R}^n$, the value of $\omega$ at $x$ on $(v_1,\dots,v_k)$ is denoted by

When $k=0$, this means exactly that a differential $0$-form on $U$ is a real-valued function on $U$.