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In this paper we construct free resolutions of certain class of closedsubvarieties of affine space of symmetric matrices (of a given size). Our classcovers the symmetric determinantal varieties (i.e., determinantal varieties inthe space of symmetric matrices), whose resolutions were first constructed byJ\'ozefiak-Pragacz-Weyman. Our approach follows the techniques developed inFree Resolutions of Some Schubert Singularities byKummini-Lakshmibai-Sastry-Seshadri, and uses the geometry of Schubertvarieties.

Free Resolutions of Some Schubert Singularities in the Lagrangian Grassmannian