Orbit Determination Question for Numerical Launcher Simulation [duplicate]

For a Senior Design Project (Which has been completed and submitted) I (and another teammate) made a numerical simulation for a single stage launch vehicle from mars that included a lot of various effects like gravity varying as a function of height, atmospheric drag (with the $C_d$ varying as a function of mach number) and the gravity turn.

We never really figure out how to calculate the orbit at each and every time step which would have been very useful. We did have the altitude and velocity at every time point. I was wondering how one would go about calculating the orbital parameters at every point.

I was thinking that because the launch location was known and the altitude, velocity and down range distance were all calculated at each time step, could I find the position and velocity vectors for each time step and work from the vectors to find the orbital parameters? or would some other method need to be be used?

In addition what is the radial vector aligned to? As in if the launch site is Eberswadale Crater (24°S, 33°W) with a altitude of 0 m (just using this as an example altitude), what would the starting radial vector and velocity vector be?