Let $Q_n$ denote a random symmetric ( $n{\times}n)$ -matrix, whose upper-diagonal entries are independent and identically distributed (i.i.d.) Bernoulli random variables (which take values $0$ and $1$ with probability $1/2$ ). We prove that $Q_n$ is nonsingular with probability $1-O(n^{-1/8+\delta})$ for any fixed $\delta > 0$ . The proof uses a quadratic version of Littlewood-Offord-type results concerning the concentration functions of random variables and can be extended for more general models of random matrices