Let $\omega = \sum_i \omega_i dx^i = \sum_{i} \nu_i dy^i$ be a 1-form in two different bases. Now, $(\omega_1,...,\omega_n)$ transform covariantly to $(\nu_1,...,\nu_n).$ My question is, can we somehow write this transformation in terms of a Jacobian of the charts?
Cause, for a contravariant trafo from $X = \sum_i a^i \partial_i $ in a chart $\phi$ to $X = \sum_i b^i \tilde{\partial_i}$ in a chart $\psi$ we have
$$(b_1,..,b_n) = D(\psi \circ \phi^{-1}) (a_1,...,a_n).$$ Is something similar possible in the covariant case, too?
Yes, the transformation is exactly the inverse of the transformation in the contravariant case. Note that $y=(\psi\circ\phi^{-1})(x)$, so $dy^i=\sum_j [D(\psi\circ\phi^{-1})]_j^i dx_j$, and the coordinates of $\omega$ transform by the inverse.