  The spin density matrix of the $\omega$ has been determined for the reaction
$\bar{p} p \, \rightarrow \, \omega \pi^0$ with unpolarized in-flight data
measured by the Crystal Barrel LEAR experiment at CERN. The two main decay
modes of the $\omega$ into $\pi^0 \gamma $ and $\pi^+ \pi^- \pi^0$ have been
separately analyzed for various $\bar{p}$ momenta between 600 and 1940 MeV/c.
The results obtained with the usual method by extracting the matrix elements
via the $\omega$ decay angular distributions and with the more sophisticated
method via a full partial wave analysis are in good agreement. A strong spin
alignment of the $\omega$ is clearly visible in this energy regime and all
individual spin density matrix elements exhibit an oscillatory dependence on
the production angle. In addition, the largest contributing orbital angular
momentum of the $\bar{p}p$ system has been identified for the different beam
momenta. It increases from $L^{max}_{\bar{p}p}$ = 2 at 600 MeV/c to
$L^{max}_{\bar{p}p}$ = 5 at 1940 MeV/c.
