The projection of the element of two-dimensional surfaces in three dimensions on coordinate planes $x_\alpha x_\beta$ is an antisymmetric tensor of rank $2$
$$df_{\alpha\beta}=dx_{\alpha} dx_{\beta}'-dx_{\beta} dx'_{\alpha}$$
It is said in a textbook that as an element of integration over the two-dimensional surfaces in three dimensions is taken the dual of $df_{\alpha\beta}$
$$\ast df_{\alpha\beta}=\frac{1}{2}\varepsilon_{\alpha}\small^{\beta\gamma}df_{\beta\gamma}$$
How to understand intuitively the reason for switching from the projection to its dual?