Expressing 3D orientation in alternative to Euler angles for 3D rigid body dynamics

I was unsure whether it would be best to post this in Physics, Maths, or other forums, so please say if this question is suited better elsewhere.

I am trying trying to create a physics engine for a game that makes makes use of Euler's equation of motion for 3D rigid body rotation. In order to do so, I needed to describe the orientation with a set of three angles. The choice I first made was using Euler angles. This seemed okay in theory, but I started to realise a problem with this whenever I tried to use Euler's equation.

Euler's equation is expressed in terms of angular velocities about the principal axes (1,2,3) of the rotating body, and these angular velocities are equal to:

$$\omega_1 = \dot\theta\sin{\psi}-\dot\phi\sin{\theta}\cos{\psi}$$ $$\omega_2 = \dot\theta\cos{\psi}-\dot\phi\sin{\theta}\sin{\psi}$$ $$\omega_3 = \dot\psi + \dot\phi\cos{\theta}$$

(Note: a z-y-z convention for the Euler angles have been adopted)

Whenever you eliminate $\dot\theta$ from the first two equations, you get:

$$\dot\phi = \frac{\omega_2 \sin{\psi} - \omega_1 \cos{\psi}}{\sin{\theta}}$$

This means for $\theta \rightarrow 0$, the precession rate, $\dot\phi$, goes to infinity, and infinity really isn't too healthy for my simulation.

So, is there an alternative angle system that bypasses this? Have I overlooked something that solves this issue? I'm sure this has been solved before as there are plenty of video games that have 3D rotation without anything crashing!