If you like clutching maps descriptions of bundles the sphere has a nice one. Think of $S^n$ as the union of two discs corresponding to an upper and lower hemi-sphere. Then the tangent bundle trivializes over both hemispheres. You can write down the trivializations explicitly with some linear algebra constructions. Think of the intersection of the two hemi-spheres as an $S^{n-1}$, this allows you to think of the tangent bundle as a union $D^n \times \mathbb R^n \cup D^n \times \mathbb R^n$ along the common boundary $S^{n-1} \times \mathbb R^n$. The clutching (gluing) map is then a map of the form:

$$ c: S^{n-1} \to SO_n $$

and it is explicitly the map $c(v) = M(v)M(x_{n+1})$

where $v \in S^{n-1} \subset \mathbb R^n \subset \mathbb R^{n+1}$ where we think of $\mathbb R^n$ as the orthogonal complement of the $(n+1)$-st coordinate vector $x_{n+1}$ in $\mathbb R^{n+1}$. $M(v)$ denotes mirror reflection fixing the orthogonal complement to $v$.

The basic idea in this construction is that if one takes a geodesic between two points on a sphere, parallel transport from one point to the other can be written as a composite of two reflections, the latter reflection corresponding to the mid-point of the geodesic, the initial reflection corresponding to the initial point of the geodesic.