Wedge product/Elementary properties/Fact

${\displaystyle V^{n}\longrightarrow \bigwedge ^{n}V,(v_{1},\ldots ,v_{n})\longmapsto v_{1}\wedge \ldots \wedge v_{n},}$

is multilinear and alternating.

${\displaystyle v_{1}\wedge \ldots \wedge v_{i-1}\wedge v\wedge w\wedge v_{i+2}\wedge \ldots \wedge v_{n}=-v_{1}\wedge \ldots \wedge v_{i-1}\wedge w\wedge v\wedge v_{i+2}\wedge \ldots \wedge v_{n}\,.}$

${\displaystyle {}u_{1},\ldots ,u_{m}\in V}$

${\displaystyle {}v_{j}=\sum _{i=1}^{m}a_{ij}u_{i}\,}$

for ${\displaystyle {}j=1,\ldots ,n}$. Then

${\displaystyle {}{\begin{aligned}v_{1}\wedge \ldots \wedge v_{n}&=\sum _{(i_{1},\ldots ,i_{n})\in \{1,\ldots ,m\}^{n}}{\left(\prod _{j=1}^{n}a_{i_{j}j}\right)}u_{i_{1}}\wedge \ldots \wedge u_{i_{n}}\\&=\sum _{1\leq i_{1}<\ldots <i_{n}\leq m}{\left(\sum _{\pi \in S_{n}}\operatorname {sgn} (\pi )\prod _{j=1}^{n}a_{i_{\pi (j)}j}\right)}u_{i_{1}}\wedge \ldots \wedge u_{i_{n}}.\end{aligned}}}$