The group $SO(n)$ has a representation in terms of infinitessimal generators on $\mathbb{R}^n$
$$M^{\mu\nu} = x^\mu \partial_\nu - x^\nu \partial_\mu$$
Does the $Spin(n)$ have a representation in terms of differential operators also? In real space (not superspace)? I suspect that it might but in a dimension higher than n.
I know the representation in terms of spinor fields but this is an infinite dimensional representation.