Spinor bundle

For the moment, what I have is this:
Given the smooth oriented Lorentzian manifold $(M,g)$, we have the tangent bundle $TM\to M$.
We also have the $\mathrm{SO}(n,1)$-orthonormal frame bundle, $P\to M$. Now, as we know, $TM$ is the bundle associated to $P$ via

(see associated bundle).
Since $\mathrm{SO}(n,1)$ has the double cover $\mathrm{Spin}(n,1)$, we suppose the existence of a "kind of double cover bundle" of $P$, i.e., another principal bundle $\mathrm{Spin}(n,1)\to Q\to M$, with a double cover map $\varphi :Q\to P$ (with some additional requirements. I think this is what is called spin structure (see also spin structure in Wikipedia). We then take a vector space $\mathrm{\Delta }$ over which there is a representation of $\mathrm{Spin}(n,1)$, this is where the Clifford algebra comes in. The spinor bundle is finally $S=Q{\times }_{\mathrm{Spin}(n,1)}\mathrm{\Delta }$. So the spinor bundle is a vector bundle associated to the double cover of the orthonormal frame bundle.