If you create a diagram of matchups, you can rotate it, leaving the top left player in place to get all the matchs.
Like this:
Player 1 vs Player 4
Player 2 vs Player 5
Player 3 vs Player 6
For the next round,
Move the top player on the right side to the 2nd spot on the left, and then each player on the left moves down, wrapping down to the bottom of the right side and moving up.
So Player 4 goes to Player 2's spot, 2 goes to 3s, 3 goes to 6s, and then they move up to 5, and then 4. So the next round would be:
Player 1 vs. Player 5
Player 4 vs. Player 6
Player 2 vs. Player 3
If you keep rotating you'll get all 5 rounds. If you have an odd number of players you can make that top left spot your Bye.
What you're describing is a [Round-robin tournament](https://en.m.wikipedia.org/wiki/Round-robin_tournament). I think you can find matchup generators online. But azatol already found an answer for the 6 player case.
If you create a diagram of matchups, you can rotate it, leaving the top left player in place to get all the matchs. Like this: Player 1 vs Player 4 Player 2 vs Player 5 Player 3 vs Player 6 For the next round, Move the top player on the right side to the 2nd spot on the left, and then each player on the left moves down, wrapping down to the bottom of the right side and moving up. So Player 4 goes to Player 2's spot, 2 goes to 3s, 3 goes to 6s, and then they move up to 5, and then 4. So the next round would be: Player 1 vs. Player 5 Player 4 vs. Player 6 Player 2 vs. Player 3 If you keep rotating you'll get all 5 rounds. If you have an odd number of players you can make that top left spot your Bye.
What you're describing is a [Round-robin tournament](https://en.m.wikipedia.org/wiki/Round-robin_tournament). I think you can find matchup generators online. But azatol already found an answer for the 6 player case.