I just tried to update my Office and it told me that it could not run without storing permanent cookies on my PC.

After allowing this it installed an ActiveX control which it stored under the name "OGACheckControl.DLL" in my "system32" directory. Since my Office passed the genuine check, I can't tell you much about how to crack the OGA. But it seems to contain much code from the WGA check, so a crack might be done in the same way.