Let’s first define what we understand by “representative democracy”. After all, remember that when the Constitution of the US was first created, slaves were 3/5 of a person and only men who had certain amount of property could vote (full national suffrage for women in the US didn’t arrive until 1920, although a few territories allowed it earlier; amendments to eliminate property requirements were introduced during the 19th century).
With that in mind, then, and if we consider that parliaments are the essential feature of anything that might approach a representative democracy…
It can be argued that The Old Swiss Confederacy might perhaps qualify, and it was founded in 1291. It was dissolved by Napoleon but, after him, Switzerland was reborn as a Federal state that purported to continue the tradition.
Then you have the old Icelandic Althing (Alþingi in Icelandic), which was a parliament that lasted from 930 till 1799, when it was suspended, to be re-instated in 1844AD. It still exists as the Icelandic parliament.
And of course, the House of Commons has existed since the 14th century, with elections taking place to send representatives to it documented from the 18th century onwards for sure (concomitant with the development of the figure of the Prime Minister from 1721 onwards).
Interestingly enough, there was a short-lived Republic of Corsica that existed between 1755 and 1769, and which extended universal suffrage to all males older than 25, without any property restrictions. It was the first place to implement universal suffrage in any form. Women also could traditionally vote on certain local issues.
These are the first examples that come to my mind. There will surely be more. I think that the Kingdom of Poland, prior to its dissolution at the hands of Prussia and Russia in the 18th century, had a very interesting system, with an elective monarchy.
Just my 2 eurocent!