Schur symmetric functions

Question

Let $s_{\lambda}$ denotes the Schur function associated to a partition $\lambda$ of $n$. Let $\lambda = (3,2)$, then by Giambelli's formula $s_{(3,2)} =det\begin{pmatrix} s_{3} & s_{4} \\ s_1 & s_2 \end{pmatrix}$. The conjugate partition of $(3,2)$ is $(2,2,1)$. So $s_{(2,2,1)}$ can be written as $s_{(2,2,1)}=det\begin{pmatrix} s_{(1,1,1)} & s_{(1,1,1,1)} \\ s_1 & s_{1,1} \end{pmatrix}$. Now the question is can $s_{(3,2)}+s_{(2,2,1)}$ be written in some nice form (determinant or pfaffian or resultant or some other) in terms of $s_{\mu}+s_{\mu'}$ where $\mu'$ is the conjugate partition.