I'm looking for a proof that if $v\_1, ..., v\_{n-1} \in \mathbb{R}^n$, then $\wedge (v\_1, ..., v\_{n-1})$ is orthogonal to each $v\_1, ..., v\_{n-1}$ for all $n \in \mathbb{N}$. 

[![enter image description here](https://i.stack.imgur.com/MwLsZ.png)](https://i.stack.imgur.com/MwLsZ.png)