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?

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?