## wedge product and change of variables

The question is:

Let $\phi: \mathbb{R}^n\rightarrow\mathbb{R}^n$ be a $C^1$ map and let $y=\phi(x)$ be the change of variables. Show that

d $y_1\wedge...\wedge$d $y_n$=(detD $\phi(x)$) $\cdot$d $x_1\wedge...\wedge$d $x_n$.

Take a look at here and the answer given by Michael Albanese:
differential geometry - wedge product and change of variables - Mathematics Stack Exchange

My question is can we prove it without using the fact " $\det A = \sum_{\sigma\in S_n}\operatorname{sign}(\sigma)\prod_{i=1}^na_{i \sigma(j)}$"?