  Baker and Bowler (2019) showed that the Grassmannian can be defined over a
tract, a ring-like structure generalizing both partial fields and hyperfields.
This notion unifies theories for matroids over partial fields, valuated
matroids, and oriented matroids. We extend Baker-Bowler theory for the
Lagrangian Grassmannian which is the set of maximal isotropic subspaces of a
$2n$-dimensional symplectic vector space. By Boege et al. (2019), the
Lagrangian Grassmannian is parameterized into the projective space of dimension
$2^{n-2}(4+\binom{n}{2})-1$ and its image is exactly the solutions of quadrics
induced by determinantal identities of principal and almost-principal minors of
a symmetric matrix. From the idea that the strong basis exchange axiom of
matroids captures the combinatorial essence of the Pl\"{u}cker relations, we
define matroid-like objects, called antisymmetric matroids, from the quadrics
for the Lagrangian Grassmannian. We also provide its cryptomorphic definition
in terms of circuits capturing the orthogonality and maximality of vectors in a
Lagrangian subspace. We define antisymmetric matroids over tracts in two
equivalent ways, which generalize both Baker-Bolwer theory and the
parameterization of the Lagrangian Grassmannian. It provides a new perspective
on the Lagrangian Grassmannian over hyperfields, especially, the tropical
Lagrangian Grassmannian. Our proof involves a homotopy theorem for graphs
associated with antisymmetric matroids, generalizing Maurer's homotopy theorem
for matroids. We also prove that if a point in the projective space satisfies
$3$-/$4$-term quadrics for the Lagrangian Grassmannian and its supports form
the bases of an antisymmetric matroid, then it satisfies all quadrics, which is
motivated by the earlier work of Tutte (1958) for matroids and linear spaces.
