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?

That the cross product has been defined as a vector at all is perhaps an accident of the fact that we perceive three spatial dimensions.  It does not make sense in any other number of dimensions (e.g. 2 or 4, though some unnatural version can be constructed in 7).  The exterior product (also called the wedge product) makes sense in any number of dimensions.  It may be envisaged as an oriented surface with a direction of rotation, the area (or magnitude) of which is equal to the parallelogram defined by the vectors.  These can be added and scaled, just like vectors.  The existence of the cross product owes its existence to the isomorphism between the space of these products and the 3-dimensional vector space.  It is simply the choice of considering these products (examples include torque, oriented area, angular momentum) to be normal vectors, though they are not (witness the fact that they are called pseudo-vectors).  Their components do not scale with a change of basis (or units) as do vectors, but rather as their square.  The need to choose of left- or right-handedness results entirely from the choice to map the product onto "normal" vectors, made possible by their isomorphism.
When the vectors are two-dimensional, the product inhabits a one dimensional space (torque in a two dimensions makes sense, so why insist on a third dimension to accommodate the normal vector?).  In four dimensions, the equivalent of torque (the oriented product of a force and the length of a lever arm) inhabits a space of six dimensions, not four.
So, the short version is: it is a convention needed to deal with the unfortunate representation of the product of vectors as normal vectors.  It would be completely (and more intuitively) avoided entirely if these products were thought of as oriented areas rather than as being classical vectors. 
