  We realize the Hopf algebra $U_{q^{-1}}(so(N))$ as an algebra of differential
operators on the quantum Euclidean space ${\bf R}_q^N$. The generators are
suitable q-deformed analogs of the angular momentum components on ordinary
${\bf R}^N$. The algebra $Fun({\bf R}_q^N)$ of functions on ${\bf R}_q^N$
splits into a direct sum of irreducible vector representations of
$U_{q^{-1}}(so(N))$; the latter are explicitly constructed as highest weight
representations.
