Invoicing question - when to send? (new at this!)

Hi all. I'm new to self-employment. I work with charities as a freelancer and consultant. The end of December will mark the end of Month 1 of my first contract (set to run for a total of three months).

I need to give them an invoice covering the first month in order to get paid. I would like to get paid as close to the end of the month as possible, seeing as I've gone from employment to contracting and so last got paid at the end of November. Of course, I know it's not a case of giving them an invoice at the end of Dec and getting paid within a couple of days. I want to get it to them ASAP to give it time to percolate through the finance department, and I know I also need to give a payment window/terms - payment within 7, 10 or 14 days, for example.

I was thinking maybe I could give them an invoice tomorrow or the next day to cover the entire month of December, with terms requiring payment within seven days of 31 December. However, this would mean submitting an invoice covering work not yet done, as I'm working next week as well as this. Do people do this? Is this what's known as a pro forma invoice? Any other suggestions??

Any guidance much appreciated....