I am quite new in the topic and I have maybe a strange question. Let $\mathbb{R}^n$ and consider a $m$ dimensional linear subspace $W$ (so $W = span(w_1, ..., w_m)$ ) of $\mathbb{R}^n$. If you want to calculate the tangent space on a particular point on $W$. How to do this?
kind regards
The tangent space at any point of $W$ is simply $W$ (at least for the most basic definition of tangent space).
If (as in Spivak's Calculus on Manifolds) your notion of a tangent space is that it's a set of pairs $(p; v)$, where $p$ is an element of your manifold, and $v$ is a vector tangent to your manifold at $p$, then your set of pairs at the point $w \in W$ is just
$$ T_wW = \{(w; u) \mid u \in W \}. $$