It's well known that for a complex vector bundle $E$, we have

$$c_n(E)=e_n(E_\mathbb{R}) $$

But I'm very curious about the relationship between the top Chern class of spinor bundle and the Euler class of tangent bundle over the same even-dimensional sphere $S^{2n}$. Why is the top Chern class of spinor bundle over $S^{2n}$ is the non-zero multiple of Euler number of $S^{2n}$?

Could you give me some help with the details? Thanks!