Can I skip the story completely after completing the game once?

Yes.

Also, map exploration and shrines are account-wide.

Finishing the campaign unlocks the campaign skip for new characters. Seasonal characters will not be able to access seasonal content (malignant hearts, extra story, etc) until the campaign is complete or skipped... so unless you feel like playing through the whole campaign again, you'll want to complete it with your eternal character first.

Oh I thought you needed to complete the game before you could even play season?

I've been spending the last couple of days hunting down Altars of Lilith. Man that took forever! lol

Thanks for all your answers. I completed the story now. But I am far from finding all altars...

From what I've heard, yes, by completing the story once on your account all (inc. seasonal) characters are able to skip the story entirely.

There are also some account-wide benefits like the shrines, and apparently they were considering making map exploration stick - though I haven't seen that confirmed.