Learn Croatian: Da, jesmo.

Smo is a short form word (a clitic) and it cannot stand alone or come right after a pause like a comma. When giving a short answer like this, you must use the full form jesmo.

Why isn't the word for 'we' (mi) in this sentence?

In Croatian, the verb form already tells you who is doing the action. Jesmo specifically means 'we are'. You could say Da, mi jesmo, but leaving the pronoun out is the most natural way to speak.

Can I just answer with Da instead of Da, jesmo?

Yes, just saying Da is perfectly fine. Replying with Da, jesmo is just like saying 'Yes, we are' instead of a plain 'Yes'. It sounds a bit more complete.

How would I say the opposite: 'No, we aren't'?

You would use the negative form of the verb for 'we', which is nismo. The complete negative answer would be Ne, nismo.