Are you aware of LinearAlgebra.issymmetric()?
afaik issymmetric() doesn’t have a way of specifying a tolerance though.
There is the build-in function isapprox, which allows you to specify absolute and relative tolerances. You could just broadcast this over the full array (A - A') and reduce with all, e.g you could define:.
isapproxsymmetric(A) = all(isapprox.(A - A', 0; rtol=1e-10))