What is the fiber of the tangent bundle?

Question

A fiber bundle is a space $(E,B,\pi,F)$ such that $\pi:E\rightarrow B$ and $E$ locally looks like the product space $B\times F$.

If M is a smooth manifold, the tangent bundle on M is the space $(T(M),M,\pi,F)$, which is also a fiber bundle. What is the fiber $F$ of the tangent bundle? the fiber at $p$ is denoted $T_p(M)$, but is there a "universal fiber?" If not, why is there a universal fiber for other fiber bundles?

Answer 1

The fiber would be the model space of the manifold, namely $\mathbb{R}^n$ if the manifold is defined to be locally diffeomorphic to $\mathbb{R}^n$, whose elements can be identified with tangent vectors.

Answer 2

The fiber of a fiber bundle is defined to be the space $F$. By definition. So the fiber of the tangent bundle of an n-dimensional manifold is $\mathbb R^n$.