Let $a_i=[a_{i1},a_{i2},\ldots,a_{in}]\in \mathbb{R}^n$, for $i=1,\ldots,n-1$. How to prove that
$ \sum_{i_1,\ldots,i_{n-1}=1}^n \varepsilon_{i,i_1,\ldots,i_{n-1}} a_{1,i_1} a_{2,i_2}\cdots a_{n-1, i_{n-1}}= (-1)^{1+i} \det \left [ \begin{array}{rrrrr} a_{11} & \ldots &\hat{a_{1i}} & \ldots & a_{1n}\\ a_{21} & \ldots & \hat{a_{i2}} & \ldots & a_{2n} \\ \ldots & \ldots & \ldots & \ldots & \ldots \\ a_{n-1,1}& \ldots & \hat{a_{n-1,i}} & \ldots & a_{n-1,n} \end{array} \right ], $
for $i=1,\ldots,n$, where $ \varepsilon_{i,i_1,\ldots,i_{n-1}}=1$ or $-1$ or $0$ depending on whether $(i,i_1,\ldots,i_{n-1})$ is an even permutation or an odd permutation or it is not a permutation of numbers $1,\ldots,n$.
(Symbol $\hat{a_{ij}}$ means that $a_{ij}$ is omitted.)
Thanks.