Is the tangent space at a point on a surface in $\mathbb{R}^3$ a subspace of $\mathbb{R}^3$?

Question

I am confused about the exact "type" (to borrow from computer science terminology) that tangent spaces have. Let's say I have a sphere $S$ in $\mathbb R^3$, and a tangent space $T_xS$ at some point $x$ on the sphere. Now, here is what I am wondering: is $T_xS$ a subspace of $\mathbb R^3$ containing 3-vectors, or is it $\mathbb R^2$, containing 2-vectors? Obviously both choices are isomorphic, but the difference is important if specific elements of the tangent space are to be written down on paper.