One of my US instructors told me, a long time ago, that if you did manage to get a PA28 taper wing into a spin (which, he said, in itself was no mean feat), unless you were in the utility category, it might spin flat and be unrecoverable, this being one of the reasons Piper never sought spinning certification.
Id be surprised, there is a strong pitch forward at the stall int eh PA28-181, I am under the belief that to flat spin you would have to have at least a neutral pitching moment at the stall.
I may be wrong because I've only done stalls in the PA28-181 with the two rear seats empty