Let $\mathfrak {g}$ be a non-degenerate triangular Lie bialgebra with the non-degenerate triangular structure $r \in \bigwedge^2 \mathfrak {g}.$ Then how does it induce $r^{-1} \in \bigwedge^2 \mathfrak {g}^{\ast}\ ?$

Since $r$ is a non-degenerate triangular structure on $\mathfrak {g}$ it induces an isomorphism $r^{\sharp} : \mathfrak {g}^{\ast} \longrightarrow \mathfrak {g}.$ So ${r^{\sharp}}^{-1} : \mathfrak {g} \longrightarrow \mathfrak {g}^{\ast}$ exists as an isomorphism of vector spaces. But how does it give rise to an element $r^{-1} \in \bigwedge^2 \mathfrak {g}^{\ast}\ $? Is it $\left ({r^{\sharp}}^{-1} \right )^{\otimes 2} (r)\ $?

Any help in this regard would be warmly appreciated. Thanks for your time.