First you have to CREATE your event. There are several ways to get to the create page, but here is one place: https://www.facebook.com/events/calendar
Click on the actual event page from the calendar (or if you have the URL handy, you can just pop that into your address ):
Under Edit, click on "Create Repeat Event" (note: you must have the event created first)
This will create a duplicate event which you'd have to do 31 times to cover a 31 day month. Ugh. 31 separate events!
Another option is to change the start and end dates to encompass the whole range. Unfortunately, this only puts it on the event calendar for the first day but you'd then have a link to send out to everyone that wouldn't be past the date.
In order to get an "end date" you have to add a time.
So, at least you have a link that isn't outdated. You could combine the two, but you'd probably want to not invite your friends to 31 events! :-)