Let $(V,Q)$ be a finite-dimensional quadratic space over either the real or complex numbers with a non-degenerate, but not necessarily positive-definite quadratic form $Q$. We define the Spin group $Spin(V,Q)$ as $Pin(V,Q) \cap Cl_0(V,Q)$, meaning as the subset of the Clifford algebra which is generated by an even number of elements $v \in V$ with $Q(v) = \pm 1$.
It is easy to prove that $$ 0 \to \mathbb{Z}_2 \to Spin(V,Q) \stackrel{Ad}\to SO(V,Q) \to 0$$ is exact, so all that remains to be proved for the adjoint representation $Ad$ to be a covering map is that it is a local homeomorphism.
We can equip both $Spin(V,Q)$ and $SO(V,Q)$ with the subspace topology induced by a norm of our choosing on $Cl(V,Q)$ and $GL(V)$, respectively. These are finite-dimensional vector spaces, so all norms induce the same topology.
Now, I do not quite know how to proceed. Could someone either point me to sources which prove that $Ad$ is a local homeomorphism or prove it themselves?