Well, quite a few societies have been matrilineal, in that men joined the women's family and property devolved along the wife's family. This mimics many social organizations in animal group (hyenas, lions, wolves, chimps) , if tribal identity is considered their "property", but the males still usually end up in dominant positions. The neurochemist in me says that the consistently higher levels of testosterone, cortisol and serotonin in the males' brains makes dominance over the females a forgone conclusion, but this is a broad generality that could be significantly varied in some populations.
In the earliest socities, political leaders tended to naturally emerge from military leaders, men with experience ordering large scale operations coordinating a large group towards a common goal. Not to mention the ability to physically dominate anyone who might challenge your authority. The most likely way women could dominate is through a shamanistic monopoly over the spiritual state of the group.
"It's a general fact that women are more rational than men in tense situations..."
Did you get taken in by Hillary's "I'll answer the red phone at 4 AM" ads?
Did Herodotus have any sort of legimtimate source on which to base the Amazons? Maybe he thinking of the Maenads, the wild drunk women who worshipped and then tore apart Dionysus?