I am reading the introductory manifolds text by Tu.
The tangent space of $\mathbb{R}^n$ at $p$, represented by $T_p(\mathbb{R}^n)$ is isomorphic to $\mathcal{D}_p(\mathbb{R}^n)$, the set of all point derivations at $p$.
Therefore, the basis $\{e_i:1\le i\le n\}$ corresponds to the set of partial derivatives $\{\frac{\partial}{\partial x^i}:1\le i\le n \}$.
So then, say we have $x(u,v)=(u,v,-\sqrt{1-u^2-v^2})$, a parametrization of the lower half of $S^2$. I don't think it is permissible to say $T_p(\mathbb{R}^n)=\{x_u,x_v\}$ since $\{x_u,x_v\}$ cannot represent any directional derivative in the $\hat{k}$ direction at any point in $\mathbb{R}^n$.
But, if I am to parametrize the lower half of $S^2$ by $x(u,v)=(\sin u\cos v,\sin u\sin v,\cos u)$, it seems plausible $\{x_u,x_v\}$ could be identified with $T_p(\mathbb{R}^n)$ as every direction is accounted for.
Is this a correct way to interpret tangent spaces?
$T_p(\Bbb R^3)$ is not $\operatorname{span}\{\frac{\partial}{\partial u}, \frac{\partial}{\partial v}\}$ because $T_p(\Bbb R^3)$ is three dimensional. However $T_p(S^2)$ is $\operatorname{span}\{\frac{\partial}{\partial u}, \frac{\partial}{\partial v}\}$ since $T_p(S^2)$ is $2$-dimensional and we have two directions in this basis (this is true for both parameterizations).
You can think about the first parameterization as taking a point on the sphere and translating it to the $uv$-plane. Think about how the tangent vectors in the $uv$-plane translate between the plane and the sphere. These vectors span a $2$-dimensional space both in the plane and on the sphere.