Wedge product/Finite-dimensional/Basis/Fact

Let ${\displaystyle {}K}$ denote a field, and let ${\displaystyle {}V}$ denote a finite-dimensional vector space of dimension ${\displaystyle {}m}$. Let ${\displaystyle {}v_{1},\ldots ,v_{m}}$ be a basis of ${\displaystyle {}V}$, and let ${\displaystyle {}n\in \mathbb {N} }$.

Then the wedge products

${\displaystyle v_{i_{1}}\wedge \ldots \wedge v_{i_{n}}{\text{ with }}1\leq i_{1}<\ldots <i_{n}\leq m}$

form a basis of ${\displaystyle {}\bigwedge ^{n}V}$.