I'm currently starting with Lie Groups, focusing on their applications in estimation (I have an engineering background). Having a look at posts like: "jacobian involving SO(3) exponential map: $\log(R\exp(m))$" is stated that the right Jacobian is given by: $$ \mathtt{J}_r(\phi) = \frac{\partial\exp([\phi]_\times)}{\partial \phi} = I - \frac{1-\cos(\Vert\phi\Vert)}{\Vert\phi\Vert^2}[\phi]_\times + \frac{\Vert\phi\Vert-\sin(\Vert\phi\Vert)}{\Vert\phi\Vert^3}[\phi]_\times^2$$
However, I'm having difficulties to understand how by using the right Jacobian $\mathtt{J}_r(\phi)$ we are able to relate an additive perturbation $\delta\phi$ in the tangent space to a multiplicative perturbation on the manifold SO(3).
Given that a first-order approximation of $\exp([\phi + \delta\phi]_\times)$, as noted in On-Manifold Preintegration for Real-Time Visual-Inertial Odometry (page 4), is given by:
$$\exp([\phi + \delta\phi]_\times) \approx \exp([\phi]_\times) \exp([\mathtt{J}_r(\phi)\delta\phi]_\times)$$
On the one hand, I don't understand how we can compute the skew matrix $[\cdot]_\times$ of $\mathtt{J}_r\delta\phi$ because I believe that $\mathtt{J}_r\delta\phi$ is already a matrix and not a vector $\in\mathbb{R}^3$.
On the other hand, given that $\mathtt{J}_r(\phi) = \frac{\partial\exp([\phi]_\times)}{\partial \phi}\to$ Doesn't this mean that ($\mathtt{J}_r(\phi)\delta\phi$) belongs to SO(3)? Following this, would the following expression be a valid way of computing the first-order approximation?: $$ \exp([\phi + \delta\phi]_\times) \approx \exp([\phi]_\times)\mathtt{J}_r(\phi)\delta\phi$$
Thanks in advance!