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Quick answer to point you in the right direction. That's the expression for an infinitesimal rotation operator (near the identity). To get a general rotation operator you exponentiate it so that $R(\hat{n},{\theta}) = e^{-i \theta \vec n \cdot \vec J}$. This is eqn 2.26 of the lecture notes you linked. The note doesn't provide motivation for this equation ...

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