Understanding the Faraday tensor I'm trying to get my head ahead understanding the Faraday 2-tensor.
I first started by thinking about how I've understood the electric and magnetic fields in electro/magnetostatics so far.  The electric field is just the force that a unit test charge would feel from a purely electric effect assume it didn't affect that field in any way.  The magnetic field is just a slightly more complicated version of that, where we have to take the velocity of the particle into account and we define the magnetic field vector at a given point as being in a direction perpendicular to the force felt by the (moving) test charge at that given point.  OK, I can understand that, I think.
But then I find some trouble in trying to figure out how a 2-tensor can describe a force, so I went off to Wikipedia to see how they define the Faraday tensor.  They define it in terms of the 4-potential, which is defined in terms of the electric and magnetic potentials, which are defined in terms of the electric and magnetic fields, which are themselves defined in terms of the electric and magnetic potentials, creating a circular definition.
So I ask you, how does one go about measuring the Faraday tensor in some region of space with some arbitrary electromagnetic field there?  I don't particularly care if the answer is impractical as I'm perfectly happy to pretend that we can test charges at every region of space to measure it.  I just need some way of getting a handle on this new idea that there is one field called the electromagnetic field that somehow explains all of the electric and magnetic effects that we'd otherwise analyze separately.
Thanks.
 A: You are looking for the Lorentz force law:
$$\frac{dp^\mu}{d\tau} = qu^\nu F^{\mu\nu}.$$
Geometrically, that says that feeding $q$ times the four-momentum $u$ into the Faraday tensor $F$ outputs the force $dp / d\tau$. So to measure $F$, you just distribute test charges everywhere, with various four-momenta, then measure the rate of change of their four-momenta.
In practice, it's hard to measure four-momentum as one object. Instead, we have detectors that can measure energy, and others that can measure momentum. Breaking the four-momentum into those pieces and taking the nonrelativistic limit, you get
$$ \frac{dE}{dt} = q u^i F^{0i}, \quad \frac{dp^i}{dt} = q(u^0 F^{0i} - u^j F^{ji}).$$
Once you plug in the components, the second equation is the usual Lorentz force law, and the first equation says that the electric field does work at rate $q\mathbf{E} \cdot \mathbf{v}$. 
The fact that we can split $F$ into components like this doesn't make it any less of a fundamental object. Splitting $F$ into electric and magnetic fields, like splitting spacetime into space and time, is just a convenience.
A: Though a tensor can be treated as a single geometric object, independent of any particular basis, the process of measurement usually requires the establishment of a definite basis, and you will then populate the components as found for that basis.
So if you pick a particular inertial reference frame, and a ground, you can start measuring at each location.  More commonly, things are filled in by theoretical means, which is much faster.
When you are done with the measurements, you have the components in a particular basis, but if you have managed to construct the complete tensor, you can freely work with it in a basis-free way, or transform it to the basis of your choice.
A: 
I'm trying to get my head ahead understanding the Faraday 2-tensor.

Good man for trying to understand it.  

I first started by thinking about how I've understood the electric and magnetic fields in electro/magnetostatics so far. The electric field is just the force that a unit test charge would feel from a purely electric effect assume it didn't affect that field in any way.

Stop and think about this for a minute. Start with a single electron. You know that it has an electromagnetic field. Not an electric field or a magnetic field, an electromagnetic field. OK, place it in space such that it is motionless in front of you. Is it moving? No. Is it subject to some force? No. So is it surrounded by a field of force? No. Now place a positron down near it. You instantly notice that the electron and the positron now move linearly towards each other. This linear "electric" force is only there when two electromagnetic fields interact. It takes two to tango. Which means this picture from Andrew Duffy's course is wrong: 

You can work out that a positron doesn't have an outward-pointing electric field, and an electron doesn't have an inward-pointing electric field. Because  this just doesn't fit with the fact that two electrons repel, two positrons repel, and an electron and positron attract. Instead you ought to know that both have an electromagnetic field, and they have the opposite chirality. The significance of this should be apparent if you've ever read The theory of molecular vortices, or seen Maxwell's picture on page 7 here: 


The magnetic field is just a slightly more complicated version of that, where we have to take the velocity of the particle into account and we define the magnetic field vector at a given point as being in a direction perpendicular to the force felt by the (moving) test charge at that given point.  OK, I can understand that, I think.

That you can. 

But then I find some trouble in trying to figure out how a 2-tensor can describe a force, so I went off to Wikipedia to see how they define the Faraday tensor. They define it in terms of the 4-potential, which is defined in terms of the electric and magnetic potentials, which are defined in terms of the electric and magnetic fields, which are themselves defined in terms of the electric and magnetic potentials, creating a circular definition.

Yes, it's circular. And there's a great deal of confusion between field and force. An electron has an electromagnetic field, and when two or more electromagnetic fields interact, the result is a linear and/or rotational force. If we only see the linear force because rotational forces balance, we talk of an electric field, if we only see the rotational force because linear forces balance, we talk of a magnetic field.   

So I ask you, how does one go about measuring the Faraday tensor in some region of space with some arbitrary electromagnetic field there? I don't particularly care if the answer is impractical as I'm perfectly happy to pretend that we can test charges at every region of space to measure it. 

You measure it with a test charge. You place it down and watch it move linearly, or you throw it and watch it move rotationally too. Try to envisage this for the electron and positron before stepping up to some big old electromagnetic field associated with some complex agglomeration of trillions of electrons and protons etc.   

I just need some way of getting a handle on this new idea that there is one field called the electromagnetic field that somehow explains all of the electric and magnetic effects that we'd otherwise analyze separately.

You can get a handle on it by reading up on the gravitomagnetic field and  spinors and positronium, and by knowing that that counter-rotating vortices attract whilst co-rotating vortices repel:
 
