Let $SO(3)$ be the rotation group of $\mathbb R^3$ and $\mathfrak s\mathfrak o(3)$ be its lie algebra of $3\times 3$ skew-symmetric matrices. Each element $A\in SO(3)$ acts in $\mathbb R^3$ via: $$v\mapsto A\cdot v$$ So given $X\in \mathfrak s\mathfrak o(3)$, the one-parameter subgroup $s\mapsto \exp(sX)$ of $X$ defines a group of transformations in $\mathbb R^3$: $$\begin{array}{rcll} g_s:&\mathbb R^3&\longrightarrow &\mathbb R^3\\ &v&\longmapsto & g_s(v)=\exp(sX)(v) \end{array}$$ I am trying to compute the tangent lift $T(g_s):T(\mathbb R^3)\cong \mathbb R^3\times \mathbb R^3\to T(\mathbb R^3)$ but I am stuck. How can I compute the differential of $g_s$? More precisely:
How can I compute the differential of $\exp(sX)$ as a rotation of $\mathbb R^3$?
Note: I am doing this in order to show that a Lagrangian in $\mathbb R^3$ that admits $\{g_s\}$ as symmetry group has the angular momentum as integral of motion (without using Noether's Theorem)