I've seen the following claim several times:
If $V$ is a vector space over $K$ with basis $\{e_1,\ldots,e_n\}$ then the basis to the kth exterior power of $V$ is given by the elements $\{e_{i_1}\wedge e_{i_2}\wedge\cdots\wedge e_{i_k} \mid 1 \le i_1 < i_2 < \cdots < i_k \le n\}$
Now, given some basic facts on exterior powers, it's not hard to show that the above set actually spans the $k$th exterior power. On the other hand, I've never seen a complete proof of the linear independence of this set. So my question is: What is the simplest way to show that $\{e_{i_1}\wedge e_{i_2}\wedge\cdots \wedge e_{i_k} \mid 1 \le i_1 < i_2 < \cdots < i_k \le n\}$ is linearly independent?