Let $e_1\dots,e_n$ be the standard basis of $\mathbf R^n$ and let $\varphi_1\dots,\varphi_n$ be the dual basis. I need to show that $\varphi_{i_1}\wedge\dots\wedge\varphi_{i_k}(v_1,\dots,v_k)$ is the determinant of the $k\times k$ minor of $(v_1,\dots, v_k)^T$ obtained by selecting columns $i_1,\dots, i_k$.
By definition, $$\varphi_{i_1}\wedge\dots\wedge\varphi_{i_k}(v_1,\dots,v_k)=\sum_{\sigma \in S_k} \operatorname{sgn}\sigma\cdot\varphi_{i_1}(v_{\sigma(1)})\dots\varphi_{i_k}(v_{\sigma(k)}),$$ but I don't know what simplifications I should do further.