Recall that the Chern character is a map $\operatorname{ch} : K(X) \to H^*(X; \mathbb{Q})$. For $X = S^{2n}$, it follows from Bott Periodicity for K theory that the image of the Chern character is actually $H^*(S^{2n}; \mathbb{Z})$ (see Proposition 6.1. of [this nice paper](https://arxiv.org/abs/1707.03883) by Konstantis and Parton). Note that on $S^{2n}$ the Chern character takes the form $\operatorname{ch}(E) = \operatorname{rank}(E) + \operatorname{ch}_n(E)$. Using identities betweeen symmetric polynomials, one can show that 

$$\operatorname{ch}_n(E) = \frac{(-1)^{n+1}c_n(E)}{(n-1)!}.$$

As $\operatorname{ch}_n(E) \in H^{2n}(S^{2n}; \mathbb{Z})$ we see that $c_n(E)$ must be divisible by $(n - 1)!$.

It turns out that there is a complex vector bundle $E \to S^{2n}$ with $c_n(E) = (n-1)!\alpha$ where $\alpha$ is a generator of $H^{2n}(S^{2n}; \mathbb{Z}) \cong \mathbb{Z}$. Blaine Lawson once told me that you can take $E$ to be the positive complex spinor bundle.

Note, one can use these observations to show that the only spheres which admit almost complex structures are $S^2$ and $S^6$. For more details, see the aforementioned paper.