Haar measure from axis-angle representation of $SO(3)$

My aim is to study some special representations of $$SO(3)$$ using characters, and to do this I need an explicit expression of the Haar measure on $$SO(3)$$. I've found some "versions" of the Haar measure on $$SO(3)$$, and the most common one seem to be the acquired by parameterizing $$SO(3)$$ using Euler angles (can be found here).

Since the characters of $$SO(3)$$ only depend on the angle from the axis-angle representation, it seems as if it would be suitable to find a Haar measure based on this parameterization. However, I don't really know where to start, and I've searched the web like a maniac.

Could someone provide some help?

The Haar measure can be explicitly calculated as follows in terms of a given parametrization $$R(\phi,\theta,\psi)$$ of $$SO(3)$$.
The first step is to define a metric thereon, which can be done by means of the Killing form: denoting by $$X$$, $$Y$$ and $$Z$$ the generators of the $$so(3)$$ algebra, $$[X,Y]=Z\,,\qquad [Y,Z]=X\,,\qquad [Z,X]=Y\,,$$ the Killing form on $$so(3)$$ is defined as the symmetric, bilinear form $$(\cdot\,,\cdot)$$ on $$so(3)$$ satisfying $$(X,X)=(Y,Y)=(Z,Z)=1\,,\qquad (X,Y)=(Y,Z)=(Z,X)=0\,.$$ In fact, one can check that this Killing form is indeed invariant under the Lie product, $$([A,B],C)+(B,[A,C])=0$$.
Now that we have a metric on the tangent space to the identity, we can define a metric at any point as follows: let $$\dot R=\frac{d}{dt}R(t)$$ be some vector, defined at the point $$R=R(t)$$; we may transport it back to the tangent space to the identity by means of the map induced by left multiplication. In this case, this simply means taking $$\Omega=R^T\dot R$$, which is indeed an element of the Lie algebra. Then, we may define a metric $$ds^2$$ by means of $$ds^2=(\Omega, \Omega)\,dt^2$$, the so-called Killing metric.
More explicitly, in the fundamental representation, we have $$(A,B)=\frac{1}{2}\mathrm{tr}(A^TB)\,$$ and $$ds^2=\frac{1}{2}\mathrm{tr}(\Omega^T\Omega)\,dt^2\,.$$ For instance, employing the Euler parametrization $$R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_z(\psi)$$ with $$\phi,\psi\in[0,2\pi)$$ and $$\theta\in[0,\pi)$$, $$R_z(\phi)=\left( \begin{matrix} \cos\phi & -\sin\phi & 0\\ \sin\phi & \cos\phi &0\\ 0 & 0 & 1 \end{matrix} \right) \,,\qquad R_y(\theta)=\left( \begin{matrix} \cos\theta & 0 & \sin\theta\\ 0 & 1 & 0\\ -\sin\theta &0 & \cos\theta \\ \end{matrix} \right)$$ you should obtain (after a somewhat lengthy calculation) $$ds^2= d\phi^2 + 2\cos\theta\, d\phi\, d\psi + d\theta^2 + d\psi^2\,.$$ The determinant of this metric, $$(\sin\theta)^2$$, provides the Haar measure/volume form by means of $$\omega = \sqrt{\mathrm{det}g}\,d\phi\wedge d\theta\wedge d\psi= \sin\theta \,d\phi \wedge d\theta\wedge d\psi.$$ Note that this measure is automatically right- and left-invariant, since so is the Killing metric.