Q:

From "Matrix" form to "Component" (tensor) form

Given

$\omega=-\eta\omega^T\eta^{-1}=-\eta\omega^T\eta$, 
where $\eta$ is the usual Minkowski metric. 

Is the following logic correct?: 
$$
{\omega^{~\mu}}_{\nu}= -{\eta_{\varepsilon\nu}}{\left(\omega^T\right)_{\sigma}}^{~\varepsilon}~\eta^{~\mu\sigma}= -{\left(\omega^T\right)^{\mu}}_{~\nu}=-{\omega^{~\nu}}_{\mu}
$$
and so
$$
{\omega^{~\mu}}_{\nu}+{\omega^{~\nu}}_{\mu}=0 . 
$$
I'm not sure I'm translating the equation into component form correctly, the manipulation of indices I'm (in principle) happier with. 
Thanks! 

A:

No, as you can see your indices don't match up.  You must have an equation of the form $A^\mu{}_\nu = B^\mu{}_\nu$. 
The correct way to do it is
$$
\omega^\mu{}_\nu = - \eta^{\mu\rho} ( \omega^T)_\rho{}^\sigma \eta_{\sigma\nu} 
$$
Note the way the indices are always adjacent since this is matrix multiplication.  Next, we use 
$$
( \omega^T)_\rho{}^\sigma = \omega^\sigma{}_\rho
$$
Then,
$$
\omega^\mu{}_\nu = - \eta^{\mu\rho} \omega^\sigma{}_\rho \eta_{\sigma\nu}  = - \omega_\nu{}^\mu
$$
Thus, the correct equation is
$$
\omega^\mu{}_\nu  + \omega_\nu{}^\mu = 0 . 
$$

