Here's an idea which I think might be expandable to a solution once some details are filled in. (I am rather tired at the moment, though, so apologies if there is a cretinous error in what follows.)
We'll do the case $n=4m-1$ where $m$ is an integer; the case $n=4m-3$ is similar.
Let $C$ be a $2m\times 2m$ matrix which has the required form. Let $A$ be the $n\times n$ matrix with $C$ in the top left corner, $1$ on the remaning $2m-1$ diagonal entries, and zero elsewhere. Let $B$ be the $n\times n$ matrix with $C$ in the bottom right corner, $1$ on the remaining $2m-1$ diagonal entries, and zero elsewhere.
$A=\left[\begin{matrix} C & 0 \\ 0 & I_{2m-1} \end{matrix} \right]\quad,\quad B= \left[\begin{matrix} I_{2m-1} & 0 \\ 0 & C \end{matrix} \right]$
Both $A$ and $B$ will be real orthogonal since $C$ is.
Consider the matrix $AB$, which being the product of real orthogonal matrices will also be orthogonal. I claim that the entries will all be $O(\sqrt{n})$ as required.
In more detail:
-- If both $i$ and $j$ are $\leq 2m-1$, then $(AB)_{ij}=A_{ij}=C_{ij}$ which is small by our choice of $C$; by symmetry, we can dispose of the case where both $i$ and $j$ are $\geq 2m+1$ in a similar way.
-- If $i\leq 2m-1$ and $j\geq 2m+1$, then on considering $\sum_r A_{ir}B_{rj}$ we see that the only nonzero contribution comes when $r\leq 2m$ and $r\geq 2m$, i.e. when $r=2m$ and so $(AB)_{ij}=A_{i,2m}B_{2m,j}$ is small.
-- If $i=2m$ or $j=2m$ then a similar analysis shows that $(AB)_{ij}$ can't be bigger than the entries of $C$ (at least up to some constant independent of $m$).
-- If $i\geq 2m+1$ and $j\leq 2m-1$ then $(AB)_{ij}=0$.
That should handle the case $n=4m-1$. The case $n=4m-3$ can be done in a similar fashion, but this time we will have extra factors of $3$ floating around since we have $3\times n$ and $n\times 3$ regions to consider, rather than just $1\times n$ and $n\times 1$ regions.
