In 3 dimensions, rotations, i.e., transformations corresponding to orthogonal $U$ with determinant 1, are generated by the (orbital) angular momentum operator $\vec{L} $ with components $L_i =-i \epsilon_{ijk} x_j \,\partial / \partial x_k $. By Euler's rotation theorem, any given such transformation can be effected by rotating around a specific axis $\vec{e} $ by an angle $\alpha $. Then, the desired rotation operator is $$ \exp \left(-i \alpha \ \vec{e} \cdot \vec{L} \right) \ . $$ In other than 3 dimensions, there isn't, of course, such an intuitive description in terms of a vector axis and an angle, but the modification is purely on the level of the rotation theorem -- once this is adapted, one will still then generate the rotations using the tensor operator $x_j \,\partial / \partial x_k $.

In 3 dimensions, rotations, i.e., transformations corresponding to orthogonal $U$ with determinant 1, are generated by the (orbital) angular momentum operator $\vec{L} $ with components $L_i =-i \epsilon_{ijk} x_j \,\partial / \partial x_k $. By Euler's rotation theorem, any given such transformation can be effected by rotating around a specific axis $\vec{e} $ by an angle $\alpha $. Then, the desired rotation operator is $$ \exp \left(-i \alpha \ \vec{e} \cdot \vec{L} \right) \ . $$ In other than 3 dimensions, there isn't, of course, such an intuitive description in terms of a vector axis and an angle, but the modification is purely on the level of the rotation theorem -- once this is adapted, one will still then generate the rotations using the antisymmetric tensor operator $x_j \,\partial / \partial x_k - x_k \,\partial / \partial x_j $.