There are of course lots of definitions and references for this, but in the same way that, on a manifold $M$,

* a Riemannian metric is a section of positive definite symmetric bilinear forms on $TM$
* or an almost complex structure is a section of $\textrm{End}(TM)$ which is everywhere
* or an orientation is a non-vanishing section of $\Lambda^m TM$ 

is 

a spin structure, a section of quadratic forms $Q$ on $TM$ (of type $(s,t)$) and a vector bundle $S$ so that $\textrm{End}(S) \simeq \textrm{C}\ell(TM,Q)$?

...or something of the like? any references where it may be stated in this fashion?