Question: We started learning about electromagnetism in physics class, and the Right Hand Rule comes in handy as seems easy to use, but I'm curious as to how it actually works. I guess it's more of a math question since I think it just involves solving the cross product of the velocity of the charge and the magnetic field. I don't know anything about cross products, but I searched some things up and it seems that the matrix is has unit vectors in it which determine the directions, so would one have to solve the whole thing to determine the direction of the force on the charge? I know it has to be perpendicular to both of the vectors but that still leaves 2 directions.

Answer: 
The formula for the force of a particle due to its magnetic field is $F = q \vec v \times \vec B$. The cross product has the property that its result is always perpendicular to both arguments.

Its direction is simply a result of how the cross product function is defined and the sign of electric charge (an electron is defined as negative).

It is important to note that the order of the arguments matters as the cross product function is not commutative; generally $ \vec A \times \vec B \neq \vec B \times \vec A$. For vectors the direction will be reversed; for matrices both sides can be wildly different.
The cross product does not have a specific direction, but either the left hand rule or right hand rule works, but it has to be applied consistantly. That's the key. So you simply drum in one rule in everywhere. 

It's kind of like driving to the left or to the right. Either works, but you have to be consistant about it.