Change of coordinates in tangent space

Question

Suppose I have a point $p \in M$, where $M$ is a differential manifold and suppose I have two parametrizations $\sigma_1 : V_1 \to M$ and $\sigma_2: V_2 \to M$ such that $p \in \sigma_1(V_1) \cap \sigma_2(V_2)$, where $V_1$ and $V_2$ are open subsets in $R^n$. Given a vector $\xi \in T_pM$, it can be writen as $\xi = \sum_{i = 1}^n \xi_{1i}\frac{\partial\sigma_1}{\partial x_i} $, if I express it according to the parametrization $\sigma_1$, or $\xi = \sum_{i = 1}^n \xi_{2i}\frac{\partial\sigma_2}{\partial x_i} $ if I express it accordding to $\sigma_2$. Given that $T_pM$ isn't dependent on the parametrization, there should be a change of coordinates in $T_pM$ such that I could calculate $\xi_{2i}(\xi_{11}, ... , \xi_{1n})$ for each $i = 1, ... , n$. How can I find this change of coordinate?