The unit vectors in polar coordinates describe the directions of increasing $\rho$ and $\phi$:
$$ \begin{align*} \hat \rho &= \cos(\phi) \hat i + \sin(\phi) \hat j \\ \hat \phi &= -\sin(\phi) \hat i + \cos(\phi) \hat j \\ \end{align*} $$
Note the above formulas give the polar unit vectors in-terms of the cartesian unit vectors $\hat i$ and $\hat j$. This somewhat bothers me, since it doesn't seem like there should be anything "fundamental" about cartesian coordinates. Why can't I start with polar coordinates and describe the cartesian unit vectors in terms of $\hat \rho$ and $\hat \phi$?
$$ \begin{align*} \hat i &= ?\ \hat \rho\ + ?\ \hat \phi \\ \hat j &= ?\ \hat \rho\ + ?\ \hat \phi \\ \end{align*} $$
I think that, given $\hat \rho$ and $\hat \phi$, (both of which depend on $\phi$ - or where I am in the plane), I can rotate clockwise by $\phi$ to obtain $\hat i$ and $\hat j$. This rotation is given (in the standard basis) by the matrix: $$R(\phi) = \begin{pmatrix} \cos \phi & \sin \phi \\ -\sin \phi & \cos \phi \end{pmatrix}.$$ So, for example, to get $\hat i$, I want to rotate $\hat \rho$ by $\phi$ radians. But if I want to apply $R(\phi)$ to $\hat \rho$, I need to express $\hat \rho$ in the standard basis again, which is what I'm trying to avoid.
