Moore determinant over $\ZZ/2$

Let \(\ZZ_2\) be the ring of integers modulo \(2\). Prove the identity in \(\ZZ_2[x_1,\ldots,x_n]\) \[ \det\!\begin{pmatrix} x_1&\cdots&x_n\\ x_1^2&\cdots&x_n^2\\ \vdots&&\vdots\\ x_1^{2^{n-1}}&\cdots&x_n^{2^{n-1}} \end{pmatrix} = \prod_{(a_1,\ldots,a_n)\ne(0,\ldots,0)}(a_1x_1+\cdots+a_nx_n), \] where \((a_1,\ldots,a_n)\) runs over all nonzero elements of \(\ZZ_2^n\).