Decomposition of functions between Banach spaces in the orthogonality equation

Let $E,F$ be Banach spaces. In the case that $F$ is reflexive we give a description for the solutions $(f,g)$ of the Banach-orthogonality equation $$\langle f(x),g(\alpha)\rangle=\langle x,\alpha\rangle\hspace{10mm}\forall x\in E,\forall \alpha\in E^*,$$ where $f:E\rightarrow F,g:E^*\rightarrow F^*$ are two maps. Our result generalizes the recent result of {\L}ukasik and W\'{o}jcik in the case that $E$ and $F$ are Hilbert spaces.