I can't find anywhere stating when I can use my rope skill to tie someone up.
I know it's obvious that if they're incapacitated I should question this but I'm wondering if, after I trip them and they're prone, I can do a rope check on them.
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I'd rule that as a no, not directly. Even though a character/monster may be prone, that doesn't mean he can't defend himself. Prone creatures may even attack in melee (with some difficulty) or cast spells if they have one empty hand.
Therefore I don't think one would be able to simple tie a prone character with a rope, at least easily.
You could probably offer the prone creature an opportunity attack (seems appropriate as the prone creature is probably armed while the rope user holds only a rope with his hands) as the rope-user attempts a grapple or melee touch attack the victim and then allow them a Use rope check to tie the target with a DC of your choice (presumably rather high or opposed by an escape artist check from the prone creature.)
In that way you do allow the player a chance of performing the desired task mid-combat against a merely prone target, with a certain amount of risk involved though, which seems perfectly fair to me considering the advantages of having someone restricted in that way.
I would argue that you cannot tie someone up after just tripping them. You would need to wrestle them into some sort of pin, and while Character A maintains the pin, Character B does the hog-tying. Not sure which version you are playing, but at least in Pathfinder, there are rules to go from a trip to a pin.
According to the RAW, the character must be pinned for 10 rounds while you tie him. See the grapple rules.
As a DM, I would probably house-rule to reduce the time if a second character maintained the pin while the first one did the tying. In a non-combat situation, I could see waiving the multiple escape checks that a character would normally get.