One way to construct vector bundles over spheres is given by the so-called clutching maps, that is maps (between topological spaces) of the form $$f : \mathbb{S}^{k-1} \rightarrow GL_n(\mathbb{R}).$$ Through those maps we can form a quotient of $(\mathbb{D}_+^{k} \times \mathbb{R}^n) \coprod (\mathbb{D}_-^{k} \times \mathbb{R}^n)$ (which I'll call by X), by identifying $(x,u) \sim (x, f_x (u))$ along the common boundary $\mathbb{S}^{k-1}$, of the corresponding disks and along with that construction we have an one-to-one correspondence $$[\mathbb{S}^{k-1}, GL_n(\mathbb{R})] \cong Vect^n(\mathbb{S}^k).$$
Now, I'm intrested in the special case where $k=n=2$. Is "trivial" from the Hairy Ball Theorem that the tangent bundle of $\mathbb{S}^{2}$ isn't trivial and we can recover it by the aforementioned construction. So, although the diffeomorhpism of $$h : T(\mathbb{S}^2) \rightarrow X,$$ seems quite reasonable, I can't find what's the inverse of it, since on equivalence classes needs some attention/subtlety which I can't find at the moment. Could you please help me out with that?
Thank you!