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Not sure if this is what you are really after, but anyway: The paper by Rudelson and Vershynin that Igor Rivin linked to contains lots of things which may be helpful for you. For example, for random $N\times n$ matrices ($N>n$) with iid Gaussian entries, there is Theorem 2.6 there which says that for the smallest singular value $s_\text{min}(A)$ it holds that $$\sqrt{N} - \sqrt{n}\leq \mathbb{E}s_\text{min}(A).$$ Also, there is a the quantitative bound $$\mathbb{P}(\sqrt{N}-\sqrt{n} - t \leq s_\text{min}(A)) \geq 1 - 2\exp(-t^2/2).$$ The case of square (but not necessarily symmetric) matrices is more difficult. There is also a probability for a lower bound on the smallest singular value: For square (subgaussian) matrices, Theorem 3.2 says that for some $C>0$, $0<c<1$ and any $\epsilon>0$ it holds that $$\mathbb{P}(s_\text{min}(A)\leq \epsilon n^{-1/2})\leq C\epsilon + c^n.$$