You can do:
julia> model = Model();
julia> a = SymmetricMatrixSpace()
SymmetricMatrixSpace()
julia> @variable(model, x[1:2,1:2], set=a)
2×2 LinearAlgebra.Symmetric{VariableRef, Matrix{VariableRef}}:
x[1,1] x[1,2]
x[1,2] x[2,2]
where it’s HermitianMatrixSpace() for Hermitian matrices. The docs mention set= in this section and SymmetricMatrixSpace here, but I don’t see anywhere they mention using it for programmatic construction, maybe that would be useful to add to the docs.