  Orthogonal designs and weighing matrices have many applications in areas such
as coding theory, cryptography, wireless networking and communication. In this
paper, we first show that if positive integer $k$ cannot be written as the sum
of three integer squares, then there does not exist any skew-symmetric weighing
matrix of order $4n$ and weight $k$, where $n$ is an odd positive integer. Then
we show that for any square $k$, there is an integer $N(k)$ such that for each
$n\ge N(k)$, there is a symmetric weighing matrix of order $n$ and weight $k$.
Moreover, we improve some of the asymptotic existence results for weighing
matrices obtained by Eades, Geramita and Seberry.
