Given a partition λ of a positive integer d, let Vλ denote the corresponding irreducible rational representation of the symmetric group Sd. When λ is a hook partition or a two-rowed partition, we explicitly describe the equivariant morphism Vλ ⊗ Vλ −→ V(d) in terms of the standard tableau basis of Vλ. We give similar descriptions for the morphism Vλ ⊗ Vλ′ −→ V(1d), as well as for the projection morphisms onto the irreducible factors of the tensor product V(d−1,1) ⊗ V(d−1,1). Our results can be interpreted as giving formulae for certain Clebsch-Gordan coefficients for the symmetric group.