Symplectic Hodge-Lepage decomposition in positive characteristic

Let LG(n,V) denote the Lagrangian Grassmannian of a symplectic vector space V of dimension 2n. What is the "correct" projective space into which LG(n,V) embeds? By correct we mean non-degeneracy: No hyperplane should contain the image of the embedding. In characteristic zero, the desired projective space is simply the subspace of the projective n-th exterior power of V for which interior multiplication with the symplectic form vanishes. In positive characteristic, it turns out that this embedding is in general degenerate. Nondegeneracy is important for construct linear error correcting codes associated with LG_n(V) over finite fields. We determine the 'correct' subspace by developing an analogue -for the case of positive characteristic- of the classical Lepage decomposition of the exterior algebra of V (used for example in the context of Monge-Ampere type PDEs).