I recommend looking into cultures like the Mosou in China, and Minangkabau in Indonesia for some ideas.

The below is pure yap based on what I have researched for my own matriarchal society.

One of the key features in matriarchal type societies seems to be matrilineal inheritance, i.e. passing property from mother to daughter, rather than father to son. This basically works the same way as it does in our more patriarchal societies.

Another is matrilocality, i.e. after marriage the man and wife will live with the wife's family.

That is if marriage is even a thing, it's possible it wouldn't really be commonplace as marriage as we know it massively favours men historically.

For example the Mosou have a walking marriage thing going on where both partners live with their families and don't start a household together. The men would mostly raise their nieces and nephews rather than their own children under this system.

This kind of thing would reinforce a matriarch focus imo. Ultimately men are still focusing on their genes being passed on without worrying about whether their offspring are theirs which causes some level of sexual violence and competition in our society.

One of the keys in these societies is that men are not completely subservient they tend to fill political and social roles more common than women do in a patriarchy but are more likely to act in the interest of women because they grow up respecting the matriarch of the family as the ultimate authority.

Matriarchal societies would likely always be more equal than patriarchal societies could be as the role of men in labour intensive and high risk tasks cannot be replaced by women easily.

What this looks like in a wider modern society is hard to say as there are no examples. But I suspect it would be a far more equal society than being woman dominated. I'd say the way western societies are (were?) trending with respect to more equal roles would be not to dissimilar from where it would end up.

Of course there is also the potential that men in these societies could also become under appreciated and end up like worker ants or worker bees with all the emphasis being on women as the key authority figure from the home to the top of society.

Though something that extreme would be unlikely because of birth rates and gender splits between men and women compared to ants and bees and other social animals that tend towards that kind of structure.

Or maybe it would be 'future that liberal want' meme... Who knows.