<A HREF="https://arxiv.org/abs/1105.2834">On the singularity of random Bernoulli matrices - novel integer partitions and lower bound expansions
</A> derives the lower bound (theorem 1)
$$\mathbb{P}\{ \text{det}(A_n) = 0\} \geq    2^{2-n} \binom{n}{2}-2^{-2n} \left(12 \binom{n}{2}^2-4 \binom{n}{2}\right),$$
which is of order $n^2 2^{1-n}$ for $n\rightarrow\infty$.