I don't have a way of making the full schedule that you are looking for. However, I can make an assignment of the 20 players to 5 tables such that everyone sits at a table with everyone else at least once, and at most twice - so the social mixing aspect is optimal. You could use this as the basis of a tournament, you just need to assign the tables of 4 to 2 teams of 2. Hope that is of some help.
(1 4 17 9) (2 12 19 18) (3 8 7 5) (11 10 6 15) (20 14 16 13)
(7 10 14 20) (12 17 2 8) (19 3 9 11) (15 4 5 16) (6 13 18 1)
(13 8 19 14) (10 9 16 17) (1 15 2 3) (4 18 11 7) (12 5 20 6)
(6 17 3 14) (18 9 15 8) (5 11 10 12) (19 20 4 1) (7 16 13 2)
(20 2 5 9) (17 19 15 7) (11 1 14 12) (16 6 8 4) (10 3 13 18)
(17 16 20 11) (9 14 18 5) (2 6 3 19) (8 7 1 10) (4 13 12 15)
(14 15 11 2) (3 20 8 4) (13 5 1 17) (9 12 7 6) (18 19 10 16)
(14 2 4 10) (16 1 12 3) (5 7 6 19) (15 18 17 20) (8 11 9 13)