The problem is "Find an explicit trivialization of the tangent bundle of $\mathbb{S}^{3}$", but I am confused because aren't all trivialization's of a tangent bundle just $\Phi(v^{i}\frac{\partial}{\partial u^{i}}|_{p})=(p,v^{1},...,v^{n})$? How can you get any more explicit?
EDIT: How hard is it to find a global trivialization of $\mathbb{S}^{3}$?
I cannot tell what your notation means.
But, what is meant by a trivialization of the tangent bundle $T\mathbb S^3$ is a diffeomorphism $F: T \mathbb{S}^3 \to \mathbb{S}^3 \times \mathbb R^3$ with the property that for each $p \in \mathbb S^3$, the function $F$ maps $T_p \mathbb{S}^3$ to $\{p\} \times \mathbb R^3$, and the composition $$T_p \mathbb{S}^3 \xrightarrow{F} \{p\} \times \mathbb R^3 \mapsto \mathbb R^3 $$ is a linear isomorphism.
The notation you have written does not have the appearance of such a function.