When expanding 2-forms in $(\mathbb {R}^3 )^{\wedge 2}$ into a basis, you can use a cyclic trick to get the “most natural” basis set.
$\omega = a_{1,2} dx^1\wedge dx^2+ a_{2,3} dx^2 \wedge dx^3 + a_{3,1} dx^3\wedge dx^1$
What is the most natural combination for higher dimensions and higher grades?