Symmetric matrix under orthogonal transformation still symmetric? is there any way to see that a symmetric matrix is still symmetric after applying an orthogonal basis transformation to it? I would say that a proof that refers to the entries of the matrix may be cumbersome, therefore I am asking here for clever ways to show this.

The important thing to note here is that the inverse of an orthogonal matrix is its transpose, so the basis change $S \leadsto O^{-1}SO$ is actually (also) $S \leadsto O^TSO$, in which form the symmetry-preserving nature of orthogonal basis changes is near-obvious:

$$(O^TSO)^T = O^TS^T(O^T)^T = O^TS^TO = O^TSO.$$