We solve a conditional functional equation of the form \[ x \perp^{\rho} y\Rightarrow f (x + y) = f (x) + f (y), \] where \(f\) is a mapping from a real normed linear space \((X, \| · \|)\) with \(\text{dim} X \geq 2\) into an abelian group \((G, +)\) and \(\perp^\rho\) is a given orthogonality relation associated to the norm.