Euler angles give us clear matrix for conversion of a vector from car reference $Fr^C$ to earth reference $Fr^E$. If $\vec V$ is a vector in different frames it is represented differently:
$$\vec V^E=L_{CE} \vec V^C$$
$$ L_{CE}^T= \begin{bmatrix} \cos\theta \cos\psi & \sin\phi \sin\theta \cos\psi-\cos\phi \sin\psi & \cos\phi \sin\theta \cos\psi+\sin\phi \sin\psi \\ \cos\theta \sin\psi & \sin\phi \sin\theta \sin\psi+\cos\phi \cos\psi & \cos\phi \sin\theta \sin\psi-\sin\phi \cos\psi \\ -\cos\theta & \sin\phi \cos\theta & \cos\phi \cos\theta \end{bmatrix} $$
Here $L_{CE}$ is a conversion matrix made of so many angles' sin and cosin. Here, I prefer to avoid those angles. Instead if axis $x$ in earth frame is $\vec e^E_x$ and in car frame $\vec e^C_x$ then is there any conversion using $\vec e^E_x$ and $\vec e^C_x$ instead of angles. Alternatively, I can ask how can I represent $L_{CE}$ using $\vec e^E_x$ and $\vec e^C_x$ instead of angles?