Why is cross product defined in the way that it is? 
$\mathbf{a}\times \mathbf{b}$ follows the right hand rule? Why not left hand rule? Why is it $a b \sin (x)$ times the perpendicular vector?  Why is $\sin (x)$ used with the vectors but $\cos(x)$ is a scalar product?

So why is cross product defined in the way that it is?
I am mainly interested in the right hand rule defintion too as it is out of reach?

There's a whole algebra ("Geometric Algebra", or Clifford algebra) behind the concept of multiplying vectors.  Look there to find out where the handedness comes from, why $ \cos \theta $ for $a \cdot b$ but $\sin \theta$ for $a \times b$, and many other gems.  Here's a great reference.
Essentially, the cross product is a vector that describes a plane.  A plane is described by two vectors (in any number of dimensions), but in $\mathbb{R}^3$ we have the handy situation that a plane can also be described by a vector.  So, in 3-d only, vector $c = a \times b$ describes the plane described by $a$ and $b$.  Since a plane has two sides, there's a two-fold arbitrariness ("orientation") that the choice of sign handles.
The $\sin$ and $\cos$ bit is slightly more complicated, but not much.  Check out the reference.
