does there exist any basis for vector space over the sphere?

Question

Let $M=S^2$ then the hairy ball theorem, states that there is no non-vanishing smooth tangent vector field on even-dimensional $n$-spheres. Hence, we can multiply any smooth vector field $v\in\Gamma(TS^2)$ by a function $f\in \mathcal{C}^\infty(S^2)$, which is zero everywhere except where v is, obtaining $fv=0$ despite $f\neq 0$ and $v\neq 0$. Therefore, there is no set of linearly independent vector fields on $S^2$ over $C^{\infty}(S^2)$-module. I think the same argument goes for the vector space over an algebraic field $K$. Moreover we can not express all the vector field as $g = fv$ where $f \in K \text{ and }v,g \in \Gamma(TS^2)$. If I am right does there exist any basis for vector space over the sphere?