# Is there a brief name for the symmetric space $SL_{2n} / Sp_{2n}$?

Let $V$ be a complex vector space of even dimension. Then the homogeneous space $SL_{2n}(V) / SP_{2n}(V)$ is known to parametrize the space of non-degenerate skew-symmetric bilinear forms on $V$.

(1) Can anyone provide a reference in the literature which refers to this space with a name that is briefer than "the space of non-degenerate skew-symmetric bilinear forms on $V$"?

(2) If there seem to be no adequate answers to question (1) in the affirmative, does anyone have any suggestions for a compact name of this space?

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II.1.1 in Audin/Lafontaine's "Holomorphic curves in symplectic geometry" (Birkhauser) refers to such a space as "the space of non-degenerate skew-symmetric bilinear forms on V". A briefer name for this space might be "the space of non-degenerate 2-forms on V"---I think it's suitable name, as is. –  J. Martel May 17 '12 at 4:41