  We investigate the holonomy group of a linear metric connection with
skew-symmetric torsion. In case of the euclidian space and a constant torsion
form this group is always semisimple. It does not preserve any non-degenerated
2-form or any spinor. Suitable integral formulas allow us to prove similar
properties in case of a compact Riemannian manifold equipped with a metric
connection of skew-symmetric torsion. On the Aloff-Wallach space N(1,1) we
construct families of connections admitting parallel spinors. Furthermore, we
investigate the geometry of these connections as well as the geometry of the
underlying Riemannian metric. Finally, we prove that any 7-dimensional
3-Sasakian manifold admits $\mathbb{P}^2$-parameter families of linear metric
connections and spinorial connections defined by 4-forms with parallel spinors.
