quantum mechanics - $SO(3)$, orbital angular momentum, vector product

I have a big confusion with group theory terminology. I know that orbital angular momentum (OAM) is $\mathrm{SO}(3)$-symmetric in 3D-space. Let's define QM orbital angular momentum (OAM) conventionally:

$$\pmb{L} = -i \pmb{r} \times \pmb{\nabla}$$

This definition can also be written using a set of $\mathrm{SO}(3)$ generators:

$$L^{\mu} = -i r_i \; S_{ij}^{\mu} \; \nabla_j$$

where $\mu = \{x,y,z\}$ for 3D space, and $S_{ij}^{\mu}$.

So... generators stand for the definition of a vector product in given space, essentially, definition of orthogonality? Or this is only in this case, I suppose, in which case why such a coincidence?

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If I proceed with this:

$$\pmb{r} e^{-iS^{\mu} \phi} \pmb{p}= \pmb{r} \cdot \pmb{p} - i \delta \phi \; \pmb{r} S^{\mu} \pmb{p} + \cdots = \mathrm{const} \; e^{- i \pmb{r} \cdot \pmb{p}} + \delta \phi L^{\mu}$$

Matter wave in zeroth order and OAM in first? Does it have any interpretation?