Given a smooth manifold U, we have a map $\wedge^2\Gamma(U,TU)\to \Gamma(U,TU)$ given by $X\wedge Y\mapsto [X,Y]$, where $TU$ denotes the tangent bundle. Is it possible to describe the map $\Gamma(U,T^\*U)\to \Gamma(U,\wedge^2 T^\*U)$ corresponding to this map. 



Dual of The Lie Bracket
To expand on Leonid's comment, if $\omega$ is a 1-form and $X,Y$ are vector fields, then
$$ d\omega(X \wedge Y) = X \omega(Y) - Y \omega(X) - \omega([X,Y]). $$
If the first two terms were not there, then one could say, as in Victor's answer, that the exterior derivative is (minus) the transpose of the Lie bracket of vector fields. The fact that the first two terms are there is symptomatic of Leonid's observation that the Lie bracket is not really a tensorial map.


