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 $J$ of $\textrm{End}(TM)$ which is everywhere an anti-involution (i.e. $J_x^2 = - \mathrm{Id}_{T_x M} $)
• 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?