Vector bundle $S=\Lambda^{\bullet}T^*M \otimes |\det TM|^{\frac{1}{2}}$ is called an spinor bundle for the bundle $TM\oplus T^*M$. it is an associated bundle to a $Spin(n,n)$-principal bundle where the fibers in this association is the spin representation which here is $$\rho:Spin(n,n)\to \operatorname{End}(\Lambda^{\bullet}T^*M)$$ I don't know why the following statement is true.
Here Since $|\det TM|^{\frac{1}{2}}$ is isomorphic to the trivial bundle, we see that there is always a choice of spin structure such that the spin bundle is (non-canonically) isomorphic to the exterior algebra $\Lambda^{\bullet} T^*M$.
Here I'm looking forward to figure this out: that it seems to be a fact that occurs to be always true about vector bundles. I think it's of no importance what an spinor bundle is here and this fact that I don't seem to find it happens to be true in general for any vector bundle. Something about how transition functions are involved in this isomorphism?
If I'm completely miss leaded here a hint will be a lot appreciated.