Skyrim doesn't officially
have multiplayer, but there is an unofficial multiplayer mod, though last I checked it was still in beta. I haven't tried it myself, so I don't know if you need to know someone who has the mod too, or if it offers a lobby and matchmaking. I'm sure it probably says somewhere on its site, here:
Home of the Skyrim Together and Fallout Together communities.
Having said that, from your post it sounds like you haven't played Skyrim, so it might be overwhelming to jump in with (what I presume is) a massive mod like Skyrim Together.
Elder Scrolls online differs from the regular Elder Scrolls installments (Skyrim, Morrowind, Oblivion, etc.) in that it's more of a traditional MMO experience, and all of the restrictions and tropes that go with that, whereas Skyrim is a lot more of an open sandbox where you get to decide how things go, and when.
If you're interested in a large singleplayer game in the vein of The Elder Scrolls, I would recommend Skyrim. But if your main interest is multiplayer, then perhaps get ESO instead. Just know that it's not "Skyrim Online." (It takes places hundreds of years before the main Elder Scrolls games, in fact.)
I hope this helps!