Why is angular momentum a cross product?

Hint:The property of any rotating object given by moment of inertia times angular velocity is called angular momentum. It is the attribute of a rotating body defined by the product of the moment of inertia and the rotating object's angular velocity. Also the angular momentum for a body is conserved for net torque zero on that body.

Complete step by step answer:
Angular momentum expresses the rotational property of the system. So, as we know the quantity which has magnitude as well as direction it is considered as a vector quantity. Similarly, the angular momentum is also a vector quantity. Also it depends on two other quantities which are the vector quantity itself. So, to get the angular momentum we need to get the vector (cross) product of these quantities.
 \[L = \vec r \times \vec p\]
Here, Angular momentum is $L$ ,
Radius is $\vec r$ and
Linear momentum is $\vec p$ .
We can see that both the quantities, radius and linear momentum of the body are vectorially multiplied to get angular momentum.
The angular momentum expresses the rotational property of the system, then it is necessary that the quantity must have both magnitude and directional determination.
But as a further definition, angular momentum is a vector because its generation is on a line perpendicular to the plane where the rotation happens.

Note: The linear momentum and the radius both are vector quantities. There are a lot of physical quantities like angular momentum which are a cross product of two vector quantities. Torque can be another example which is a cross product of the radius vector and the force vector acting on the body with respect to the point.