basis for exterior product

Given $\{v_1,\dots ,v_n\}$ a basis for the vector space $V$,

$$v_1\wedge v_2=-v_2\wedge v_1$$

So by this,

$$v\wedge v=-(v\wedge v)⟹v\wedge v=0.$$

To make a basis for the exterior product, ${\bigwedge }^k(V)$, want no repeating vectors and none out of order (or else there would be redundancy in the basis)

The basis for ${\bigwedge }^k(V)$ is

$$\{v_{i_1},\dots ,v_{i_k}|1\le i_1<\cdots <i_k\le n\}$$

Therefore,

$$\dim ({\bigwedge }^{k}V)=\left(\genfrac{}{}{0ex}{}{n}{k}\right).$$

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References