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We establish an explicit Plancherel decomposition for$\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$ where $E/F$ is a quadraticextension of local fields of characteristic zero by making use of a localfunctional equation for Asai $\gamma$-factors. We also give two applications ofthis Plancherel formula: first to the global Ichino-Ikeda conjecture forunitary groups by completing a comparison between local relative charactersthat was left open by W. Zhang and secondly to the Hiraga-Ichino-Ikedaconjecture on formal degrees in the case of unitary groups.

Plancherel formula for $\mathrm{GL}_n(F)\backslash \mathrm{GL}_n(E)$ and  applications to the Ichino-Ikeda and formal degree conjectures for unitary  groups