I was reading a book on theoretical quantum mechanics and the authors introduced the (orbital) angular momentum operator as the operator that generates rotations around an (arbitrary) axis. To do this, they first showed that any rotational matrix corresponding to a rotation of angle theta around an axis e can be written as: $$ \mathbf R\left(\mathbf e, \,\vartheta\right) = \exp\left[\vartheta\boldsymbol\Omega_e\right], $$
where $\Omega$ is a skew-symmetric matrix.
They then proceeded by examining how a unitary operator corresponding to a rotation acts on the wave function and eventually related the two using the formula:
\begin{align} \psi\left(e^{-\vartheta\boldsymbol{\Omega}_e}\mathbf x\right)&=\left(\hat{I}-\vartheta\boldsymbol\Omega_e\mathbf x\cdot\nabla+\frac{\vartheta^2}{2!}\left(\boldsymbol\Omega_e\mathbf x\cdot\nabla\right)^2\right)\psi\left(\mathbf x\right)+\cdots \\ &=\left(e^{-\vartheta(\boldsymbol\Omega_e\mathbf x)\cdot\nabla}\psi\right)\left(\mathbf x\right).\tag{27.82} \end{align}
which they asked the reader to prove. I've been trying to prove the result (by simply differentiating the left hand side of the equation with respect to theta) for quite a while now, but I'm unfortunately not really getting anywhere. Maybe it's just a simple trick that I'm missing.
Anyway, I'd be grateful for any suggestions or hints.