Let $A$ be an $n\times n$ random symmetric matrix with independent identically distributed subgaussian entries of unit variance. We prove the following large deviation inequality for the rank of $A$: for all $1\leq k\leq c\sqrt{n}$, $$\mathbb{P}(\operatorname{Rank}(A)\geq n-k)\geq 1-\exp(-c'kn),$$ for some fixed constants $c,c'>0$. A similar large deviation inequality is proven for the rank of the adjacency matrix of dense Erdos-Renyi graphs. This corank estimate enhances the recent breakthrough of Campos, Jensen, Michelen and Sahasrabudhe that the singularity probability of a random symmetric matrix is exponentially small, and echoes a large deviation inequality of Mark Rudelson for the rank of a random matrix with independent entries.