With 24 players you can play upto 7 rounds where all 7 partners and all 14 opponents are different,
there is a schedule here. You can assign partners and opponents in any way you like.
There are some more euchre schedules that you may be interested to see if you follow the links
here.
A solution for your problem above is possible - all 10 partners different, all 20 opponents different. But note that players 21 to 24 never sit down at a table together.
(18 11 v 6 21) ( 8 15 v 24 7) (17 10 v 19 2) (4 20 v 9 13) ( 5 23 v 12 16) (14 1 v 3 22)
(19 12 v 7 21) ( 9 11 v 24 8) (18 6 v 20 3) (5 16 v 10 14) ( 1 23 v 13 17) (15 2 v 4 22)
(20 13 v 8 21) (10 12 v 24 9) (19 7 v 16 4) (1 17 v 6 15) ( 2 23 v 14 18) (11 3 v 5 22)
(16 14 v 9 21) ( 6 13 v 24 10) (20 8 v 17 5) (2 18 v 7 11) ( 3 23 v 15 19) (12 4 v 1 22)
(17 15 v 10 21) ( 7 14 v 24 6) (16 9 v 18 1) (3 19 v 8 12) ( 4 23 v 11 20) (13 5 v 2 22)
(18 19 v 13 15) (17 7 v 22 9) (16 24 v 2 20) (4 8 v 14 23) (10 11 v 3 12) ( 1 6 v 21 5)
(19 20 v 14 11) (18 8 v 22 10) (17 24 v 3 16) (5 9 v 15 23) ( 6 12 v 4 13) ( 2 7 v 21 1)
(20 16 v 15 12) (19 9 v 22 6) (18 24 v 4 17) (1 10 v 11 23) ( 7 13 v 5 14) ( 3 8 v 21 2)
(16 17 v 11 13) (20 10 v 22 7) (19 24 v 5 18) (2 6 v 12 23) ( 8 14 v 1 15) ( 4 9 v 21 3)
(17 18 v 12 14) (16 6 v 22 8) (20 24 v 1 19) (3 7 v 13 23) ( 9 15 v 2 11) ( 5 10 v 21 4)
Hope that helps.
