Why the direction of the omega (angular velocity vector) is along the axis of rotation? Also for angular acceleration I know that the direction of omega is taken along the axis of rotation but I don't understand it why it is taken?

I also know that $\mathbf v = \mathbf ω \times \mathbf r$ so, all three vectors should be perpendicular, but this also doesn't satisfy me. As this is merely a formula but a formula don't give me that feel how that direction of omega(angular velocity vector) changes angular displacement of a body doing circular motion as this is also known that [Bold letters are vectors]
$$ \boldsymbol{\omega} = \frac{\Delta\boldsymbol{\theta}}{\Delta t} $$
Just like velocity changes linear displacement of a body.
 A: We can define the angle as the area sweep by a vector in the rotating plane with its strating point at the rotating position. As the vector rotates a angle $\Delta \theta$, the area swept by the vector is:
$$
   \Delta A = r^2  \Delta \theta
$$
Therefore, we may define the angle as the area swept divided by $r^2$.  The area is a vector quantity with direction on the normal direction of the area. This defines the direction of the angle $\omega$. I will show that the definition of area is more general that the "angle" itself.
Case 1: Consider that the rotating vector is not ristricted in the plane, e.g. it rotates in a corrugated surface. The resultant rotation angle $\theta$ won't be the sum of the infinitismal angle $\Delta \theta$
$$
  \theta  \neq \sum_i \Delta \theta_i 
$$
But it is equal to the sum of the infinitesmal area.
$$
   \theta \text{ } \hat{\theta} = \sum_i \frac{\Delta \vec{A}_i}{r^2}
$$
The $\hat{\theta}$ denotes the result vector direction of the vector summation.
Case 2: A solid angle $\Omega$ can only be defined using area
$$
  \Omega = \int \frac{\hat{r} \cdot d\vec{A}}{r^2}
$$
Hope this helps.
