Coordinate Expansion of Differential Forms on Euclidean Open Sets

Let $n\in\mathbb{N}$, let $U\subseteq \mathbb{R}^n$ be open, let $k\in\mathbb{N}\cup\{0\}$, and let $\omega$ be a differential $k$-form on $U$. Then there exist unique real-valued functions

$$a_{i_1\dots i_k}:U\to\mathbb{R}$$

indexed by strictly increasing $k$-tuples $(i_1,\dots,i_k)$ with $1\le i_1<\cdots<i_k\le n$ such that

$$\omega=\sum_{1\le i_1<\cdots<i_k\le n} a_{i_1\dots i_k}\, dx_{i_1}\wedge\cdots\wedge dx_{i_k},$$

where each $dx_i$ is the coordinate $1$-form from Coordinate 1-Form on an Open Subset of Euclidean Space and the wedge product is the one from Wedge Product of Differential Forms on Euclidean Space. When $k=0$, this says that $\omega$ is uniquely equal to a real-valued function on $U$.