Let $n,k\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be \reftext{def:open-subset-euclidean-space-2026a}{open}, let $1\le i_1<\cdots<i_k\le n$, and let $\sigma\in S_k$ be a permutation in the sense of \ref{def:permutation-initial-segment-2026a}. Then the coordinate $1$-forms from \ref{def:coordinate-1-form-euclidean-open-set-2026a} satisfy

where $\operatorname{sgn}(\sigma)$ is the sign from \ref{def:sign-permutation-2026a} and the wedge products are taken in the sense of \ref{def:wedge-product-differential-forms-euclidean-2026b}.