Tangent spaces and $\mathbb{R}^n$ The tangent space of a circle is a line.
The tangent space of a sphere (in every point) can be thought of as a plane. 
Is this a general thing? I mean, having an $n$ dimensional Riemannian manifold, can the tangent space in every point be thought as $\mathbb{R}^n$? 
If the answer is yes, does this happen as well with the Lorentzian manifolds of GR? Can the tangent space of any space-time always be regarded as a Minkowski space?

If $M$ is a smooth $n$-dimensional manifold, then for each $p\in M$ the tangent space $T_p M$ is an $n$-dimensional real vector space. This tangent space is therefore isomorphic to $\mathbb R^n$ as a real vector space, though not in a "natural" way, in the sense that $T_pM$ does not have a distinguished basis corresponding to the usual basis of $\mathbb R^n$.
This is true for any smooth manifold, regardless of whether it is equipped with a Riemannian metric or a Lorentzian metric.
