I'm trying to find a basis to the alterned tensor in $\mathbb{R}^{4}$:
$$h(x,y)=(x_{1})^{3}(y_{2})^{3}-(x_{2})^{3}(y_{1})^{3}. $$
For the tensor $f(x,y)=x_{1}y_{2}-x_{2}y_{1}$ I could write $f(x,y)=det\left[\begin{array}{cc}x_{1} \,\,\,\,x_{2} \\y_{1}\,\,\,\,y_{2} \end{array}\right]$. But for $h$ I dont know.