That really sounds like it could be a scam more than a job opportunity, since they already had you doing some real work for free, as part of the interview. Or perhaps, as a comment notes, the 'come in for training' is the job offer.
If they haven't mention what the pay rate is, then one way to approach this is to ask about pay:
I'm excited about this job but do have questions. How much are you paying, and is the training pay at the same level as the job pay? Does it just get wrapped up in the first paycheck?
In other words, act as if of course they are paying you, and you are just interested in the details. Because, if it is a real job, then of course, they are paying you for the training too. If it's a scam, they'll be evasive and not give clear answers. That is also an answer.
Since you also want to know if you've actually got the job, you can wrap that up in your questions about pay and training. Ask when your first day is after the training is done - the next day, the following Monday, something else?
In other words, sound like you assume they are offering the training as part of an offer, which of course would be paid. But, have your eyes open, and expect that it could be a scam, that they just want a free day of work from you and there will be no job offer. If you are in the US, do not do the training without filling out a W-4 form, which, of course, if they are legit, they will be planning on doing anyway.
When you suspect something is underhanded, but don't know for sure, asking questions in a manner that sounds like you are just confused but of course they would be doing the right thing -- that's not an attack if they are just a bit disorganized. And if they are trying a scam, you're asking questions instead of walking into it.