For what $n$ does there exist a self-complementary $(2n,4n-2,2n-1,n,n-1)$ balanced incomplete block design?

(All I know is that a self-complementary design with these parameters does exist for all $n$ of the form $2^k$ but doesn't exist for $n=3$.)

Motivating problem: One wants to divide $2n$ players into 2 equal-sized teams in $2n-1$ different ways in such a way that each player is on the same team as each other player exactly $n-1$ times.