Consider the following $k$-forms $\omega_k$ in $U \subset \mathbb{R}^2$ (for $U$ open): \begin{align} &\omega_0 = f\\ &\omega_1 := \langle u,\cdot \rangle = \det(u^\times,\cdot)\\ &\omega_2 := f \cdot \det \end{align}
Find relations between $u,u^\times$ and $f$, if $d\omega_0 = \omega_1$, and if $d\omega_1 = \omega_2$.
Remark: $d\omega_k$ means the exterior derivative, which we defined as
$$d(fdx^I) = df \wedge dx^I $$
, where $x^1,\ldots,x^m$ are local coordinates and $df$ is the differential of $f$.
What I do not understand here is what $u^\times$ is supposed to mean here, and I can not find this in my notes either. Does anyone happen to know what this is supposed to mean?