Can we write "Normal force" as a function of underlying surface's intrinsic properties? We usually tend to use Newton's second law for finding equations relating the forces being applied on the (not rotating) rigid body located at a plane surface and then calculate "Normal force" by extracting its value from the value of other present forces. And also we always say that the "Normal force" just depends on the surface's intrinsic properties.
Now my question is:
Can we describe/define "Normal force" as a function of surface's intrinsic physical and chemical properties such as its material type, intermolecular interactions, etc? And if we can what is the formula of this function?

EDIT(Question clarification):
By talking more with the people answered me it turned out that the "Question statement" is actually wrong.
The conclusion of our conversation is as follows:


*

*"Normal force" in general isn't a function of Body's and Surface's intrinsic properties and just equals to the opposite force the "Body" is exerting on the surface and vice-versa for the force the "Surface" is exerting on the body.

*If the Body and Surface are rigid(in its ideal sense), there is no limit to the "Normal force" they are exerting on each other since the body/surface couldn't deform and as they havn't any elastic properties, we could just say the "Normal force" is generated merely for body/surface to resist deformation.

*If the Body and Surface are elastic materials, still the normal force they are exerting to each other is equal to the opposite force exerted on both of them. But now there could be a maximum limit for "Normal force". The "Maximum Normal force" is defined the "Normal force" exerted on the body/surface when the body/surface couldn't resist deformation/peneteration anymore. Now this new quantity(Maximum normal force) is a function of both elastic body's and elastic surface's intrinsic properties. These properties are quantities we define in "Theory of elasticity", "Continuum mechanics" and "Fluid mechanics" and the function would be so complicated and will be possibly different at different situations.
So all of the following answers are correct.
For further information, see the discussion took place into the comments.
 A: You can calculate the amount of the normal force if you know the elastic properties of the material and how much it is bent or deformed out of shape. In most cases, it is very difficult to measure the amount of deformation - eg it might be less than 1 micron. Also the shape and area of the deformation have to be taken into account - except for very simple cases this can be very difficult to predict. Complex materials like wood can bend differently depending on the direction in which the force is applied. 
At its simplest this is just like stretching or compressing a spring. If you balance a weight on top of a spring, you can predict the reaction force from the spring by measuring the distance by which it is compressed, and knowing its spring constant. But if the weight comes to rest, it is simpler to realise that the forces on the weight are then in equilibrium, so the "normal reaction" from the spring is equal to the force of gravity on it. 
It is possible to derive the force constant for a helical spring from its geometry and its Shear  and Young's Moduli, or for the bending of a beam from its Young's Modulus. Both these structures are ideal models with simple loads, but the calculation is difficult enough. For real materials with complex loads, the calculation the far harder. Despite the complexity, Newton's 3rd Law still holds. 


Does the Normal Force exerted from the Surface to the Body just depend on the amount of the Force that the Body exerts on the Surface? Or does it depend on other things too? (just in the rigid body model)

The Normal Force which the Surface exerts on the Body is always equal to the Force which the Body exerts on the Surface (Newton's 3rd Law). This applies to both rigid and elastic or "soft" bodies.
Forces are interactions. They do not exist until there is some kind of interaction - either "action at a distance" such as gravity and electromagnetism, or contact forces such as air pressure, drag or dry friction. Then the bodies which are interacting exert equal and opposite forces on each other. So you cannot have a Body exerting a force on empty space, or exerting a force which does not have its "action-reaction" paired force. 
You ask "Does [the Normal Force] depend on other things too?" Yes, it depends on the acceleration of the surface and body. If the surface and body are accelerating up or down in a lift, the normal force is greater or less than the gravitational force pulling the object down towards the Earth. But Newton's 3rd Law still applies to the interacting forces between the surface and body : they are still equal and opposite. 
See Elevator normal force and If $F=ma$, how can we experience both gravity and a normal force even though we are not accelerating? and similar questions in the "Related" column and their links.
A: Think of normal force as a holding-up force. A book on a table is being held up by the table. The table can increase its holding-up force if it has to. If I push down on top of the book for example, the table must now hold back against more force. But if that holding-up force has to be larger than what the table can bear, the table breaks. 

Can we describe/define "Normal force" as a function of surface's intrinsic physical and chemical properties such as its material type, intermolecular interactions, etc?

No.
If I put a feather and a stone on a table, the table applies two different normal forces to hold these items up. But it's the same material. If you do not know the circumstances (if you don't know, which other forces that are acting), you can't know the normal force. It can be anything from 0 up to the maximum.

And also we always say that the "Normal force" just depends on the surface's intrinsic properties.

Who is the "we", who say that? It is incorrect.
The normal force is not only depending on the intrinsic properties. But the maximum normal force is. Because that just depends on the material strength.
A: Unsure what you are asking. A normal force is there to enforce a constraint. If a body cannot penetrate a surface then a force will be created in the direction that does not produce work to keep the body from interpenetrating the surface.
If it is flat (or semi-flat) surface the allowed motion directions are planar to the contact, and the normal force will be normal to the plane.
The key is that the direction of the normal force is decided by the geometry and the magnitude by the balance of forces.
Consider a ball sliding or rolling on the surface. If the point of contact has tangential velocity $\vec{v} = v \hat{x} + u \hat{y}$ then the normal force will along the direction $\hat{z}$ where $$\hat{z} \cdot \hat{x} = 0$$ $$\hat{z} \cdot \hat{y} = 0$$
This ensures the normal force will not add or remove power to the system.
A: There are are still only four known forces in the universe.  All of the other names of forces are labels for familiar manifestations of those.  The normal force is a manifestation of the electrostatic repulsion between electrons.  There is no other candidate.
On the miceo scale, when two surfaces approach each other their respective near-surface electrons and their associated atoms or molecules are displaced against restorative forces.  The net result is deformation on the macro scale.  It's not possible to model it mathematically in terms of the particle interactions because there are so many such interactions involved, not to mention the surface irregularities at the Mico level.
Rigid bodies with forces between them but no deformation are a sometimes useful fiction.
