  We consider self-propelled rigid-bodies interacting through local
body-attitude alignment modelled by stochastic differential equations. We
derive a hydrodynamic model of this system at large spatio-temporal scales and
particle numbers in any dimension $n \geq 3$. This goal was already achieved in
dimension $n=3$, or in any dimension $n \geq 3$ for a different system
involving jump processes. However, the present work corresponds to huge
conceptual and technical gaps compared with earlier ones. The key difficulty is
to determine an auxiliary but essential object, the generalized collision
invariant. We achieve this aim by using the geometrical structure of the
rotation group, namely, its maximal torus, Cartan subalgebra and Weyl group as
well as other concepts of representation theory and Weyl's integration formula.
The resulting hydrodynamic model appears as a hyperbolic system whose
coefficients depend on the generalized collision invariant.
