Let $E \to X$ be a Lie algebroid over the manifold $X$. Let $x_1,...x_n$ be local coordinates on $X$ and $e_1,...e_m$ be the basis of local sections of $E$. In terms of these coordinate functions Lie bracket and the anchor map $\rho$ are described like this: $$ [e_i,e_j]_E = \sum\limits_k c_{ijk}e_k $$ $$ \rho(e_i) = \sum\limits_j b_{ij}\frac{\partial}{\partial x_j} $$

Let $\xi_1,..\xi_m$ be the basis dual to $e_1,...e_m$. Now we define Poisson structure on $E^*$ by setting: $$ \{x_i,x_j\} = 0 $$ $$ \{\xi_i,\xi_j\} = \sum\limits_k c_{ijk}\xi_k $$ $$ \{\xi_i, x_j\} = -b_{ij} $$

This Poisson structure is independent of the choice of local coordinates and basis of local sections on $E$, so it's very possible that there exists a coordinate free expression of the Poisson structure like the one we have for dual Lie algebras. How does this expression look like?