Hi There
Nothing is showing up online for Richard & Mary other than what myself and Dathai posted.
1. The surname is a tricky one so it there may be records showing on a variation that I haven't tried or thought of.
2. There seems a big gap between when Richard & Mary married and to when Patrick was born, a not unheard of gap but just worth noting.
3. For the 1880's in Ireland you CAN get civil birth certs but you need reference numbers to get them. The problem that you are up against is that while it's a rarish surname there are still quite a few Patrick Skelly's born in and around the years that you are working with. There are also a number of MALE Skelly's too. All in all I saw about 10 possibilites just with the surname Skelly; I didn't count the other surname variations. To get research civil records cost 4 euro each - so multiply that by the 10 possibilites and you can see how expensive it can get very quickly, AND that's not to say that any of them would be for YOUR Patrick. Civil Registration was still in it's infancy here at that time and there were numerous records that were not registered and Patrick may have been one of those.
4. There are a few Dublin churches which have their old baptisms and marriages on a site called IFHF pay per view. Using their free index though I got zero matches.
Sorry not to have a better answer for you.
Tara