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Is there a minimal local representation for $SO(3)$ such that if $(x_1,x_2,x_3)$ is the representation for some $R\in SO(3)$ then I can write the entries of the 3x3 rotation matrix for $R$ as a polynomial in $x_1,x_2,x_3$?

By "minimal local representation" I mean that the representation should have dimension 3 and should be valid in some neighborhood of the identity.

By "polynomial-form" I mean that I should be able to write each of the 9 elements of the rotation matrix for $R$ as

$$ r_{ij}=Poly(x_1,x_2,x_3) $$

It would also suffice if I could write the result of rotating $u\in \mathcal{R}^3$ by the rotation represented by $x$ as

$$ Ru = Poly(u_1,u_2,u_3,x_1,x_2,x_3) $$

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