I know that the angular velocity vector $\vec{\omega}$ points towards the axis of the circular motion and there is a formula $$\vec{v}=\vec{\omega} \times \vec{r}$$
I want to know the origin of this formula which governs the direction of $\vec{\omega}$.
And why only this direction, why not the opposite direction?
Is it purely convention or is there some logic behind this?
We use a right handed coordinate system. This means that we draw our coordinate systems so that $\hat x\times\hat y=\hat z$ (described by whatever right hand rule you've been taught). The choice of right handed or left handed coordinate system is arbitrary, but once it's established as convention, you need to stick with it and not switch conventions unless you want a real headache of conversion factors trying to interpret others' work.
The axis of rotation needs to point parallel to $\overrightarrow\omega$, that's part of the definition of $\overrightarrow\omega$. $\overrightarrow\omega$ describes a right handed rotation, so it must result from the cross product of the position and velocity $\hat\omega=\hat r\times\hat v$ (note the hats, these are unit vectors). You can rotate the unit vectors to get them in the same order you had them written and multiply by their magnitudes to get your equation back.
