If the dot product corresponds to the duality bracket, the cross product corresponds to...

Question

If I consider a manifold $Q$ with some chart $(q^i)$ and if I consider the induced charts $(q^i,p_i)$ on the cotangent bundle $T^*Q$. If I consider a covector $\xi \in T^*_qQ$ and a tangent vector $v \in T_qQ$, then the duality bracket $\langle \xi, v\rangle$ corresponds to the scalar product $\xi \cdot v=\xi_1 v^1+\ldots +\xi_n v^n$, if we express $\xi=\xi_1dq^1+\ldots\xi_n dq^n$ and $v = v^1\frac{\partial}{\partial q^1}+\ldots v^n\frac{\partial}{\partial q^n}$. So we know that the dot product correspond to the duality bracket. To what operation would the cross product correspond?