$V=IR$, isn't it missing something? So I know that $V=IR$ works for circuits, but for the case of an arc-before the arc jumps, there is a potential difference, but no current, but there isn't infinite resistance is there?
I don't understand how to compute a finite resistance for an arc that would come out as infinite in some other cases.
 A: 
I don't understand how to compute a finite resistance for an arc that would come out as infinite in some other cases.

Arc formation is a sufficiently non-stationary and nonlinear process. So, one has to use dynamic circuit theory, where the resistivity in Ohm's law is a complex number and contains both active and reactive components depending on the applied voltage.
That is in general. In practice, modeling the arc's resistivity in both static and dynamic (transient) regimes is very hard problem which was attempted to be solved by many groups. Searching in Google you can find several approaches based on the equivalent circuit method, where conducting chanel is approximated by a set of resistors, capacitors and inductors. Understanding of this phenomena is strongly related to microscopic nature of electron transport in conducting channel.
A: Ohm's law in the circuitry sense can be derived from the electromagnetic sense from the equation
$$\vec J=\sigma\vec E$$
That is, current density is the conductivity times the electric field [with current density as the analog to current, conductivity the analog to the inverse of the resistance, and the electric field as the analog to voltage]. But this equation is only true for certain materials, in particular those that have sufficient free electrons. Air isn't loaded up with them so air is particularly non-ohmic [that is, it doesn't follow $V=IR$]. As an example, we know that non-zero electric field often leads to no current [when a charge is built up before the current starts flowing]. This graphic from wiki is useful [the two on the left are ohmic, the two on the right aren't]. http://en.wikipedia.org/wiki/File:FourIVcurves.svg
