Permutation Rule for Wedge Products of Coordinate 1-Forms

Let $n,k\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be open, let $1\le i_1<\cdots<i_k\le n$, and let $\sigma\in S_k$ be a permutation in the sense of Permutation of the Set $\{1,\dots,r\}$. Then the coordinate $1$-forms from Coordinate 1-Form on an Open Subset of Euclidean Space satisfy

$$dx_{i_{\sigma(1)}}\wedge\cdots\wedge dx_{i_{\sigma(k)}}=\operatorname{sgn}(\sigma)\,dx_{i_1}\wedge\cdots\wedge dx_{i_k},$$

where $\operatorname{sgn}(\sigma)$ is the sign from Sign of a Permutation and the wedge products are taken in the sense of Wedge Product of Differential Forms on Euclidean Space.