In Barrett O'Neill's book, here is the question about the wedge product of two differentials.
Let $f$ and $g$ be real-valued functions on $\mathbb{R}^2$. Prove that $$df\wedge dg= \begin{vmatrix} \frac{\partial f}{\partial x} & \frac{\partial f}{\partial y} \\ \frac{\partial g}{\partial x} & \frac{\partial g}{\partial y} \\ \end{vmatrix}dxdy$$
Then, show that it implies the alternation rule ($dx\wedge dy=-dx\wedge dy$)
I can show the above equality easily if I use the properties that $dxdx=0$ and $dydy=0$. But it seems like both of these properties were derived from the alternation rule. So, I guess there might be another way to prove the above equality. Is there any hint to prove the above equality that does not use those properties?