Write $w$ out as $a_1 e_1\wedge e_2 + a_2e_1\wedge e_3+ \cdots + a_6e_3\wedge e_4$, and then since $w\wedge y =0 $ for every $y\in W$ you can consider the four vectors taking the place of $y$: $y_1e_1\wedge e_2,\cdots y_6 e_3\wedge e_4$, and wedge these with your original thing, taking $y_i$ to be whatever you want.
In each case you'll get one term that doesn't vanish, which will be $(-1)^{\sigma(i,j)} a_jy_i e_1\wedge e_2\wedge e_3\wedge e_4$