Basis of alternating tensors

I am reading the book Calculus on Manifolds by Spivak and I am trying to solve problem 4.1(b):

Let $e_1, \dots, e_n$ be the usual basis on $\mathbb{R}^n$ and let $\varphi_1, \dots, \varphi_n$ be the dual basis. Show that $\varphi_{i_1}, \wedge \cdots \wedge \varphi_{i_k}(v_1, \dots, v_k)$ is the determinant of the $k \times k$ minor of $(v_1 \, \cdots \, v_k)^T$ obtained by selecting columns $i_1, \dots, i_k$.

Let $$v_r = \sum_{s = 1}^n \alpha_{rs} e_s.$$Then\begin{align*}\varphi_{i_1} \wedge \cdots \wedge \varphi_{i_k}(v_1, \dots, v_k) &= \sum_{\sigma \in S^k} \mbox{sgn } \sigma [\varphi_{i_1}(v_{\sigma(1)}) \cdots \varphi_{i_k}(v_{\sigma(k)})]\\&= \sum_{\sigma \in S^k} \mbox{sgn } \sigma [ \alpha_{\sigma(1), i_1} \cdots \alpha_{\sigma(k), i_k}].\end{align*}

How should I relate the determinant to this expression? Thank you