A formal synonym for fill in depends on intended sense, as fill in is informal phrasal verb. When fill in means to provide information or update someone, formal synonyms include brief, inform, apprise, advise, update, enlighten, bring up to date, and provide details. For example, in business email instead of let me fill you in, more formal would be allow me to brief you or allow me to apprise you of developments. When fill in means to substitute for someone, formal synonyms include substitute for, act as substitute for, serve as replacement for, assume duties for, act in place of, and provide coverage for. When fill in means to complete a form, formal synonyms include complete, provide information for, supply, and execute. For example, please fill in the form can be formalized as please complete the form. For adding material to fill gap, formal synonyms include complete, supplement, insert, and provide additional information. In formal writing, avoid phrasal verb fill in and use brief, apprise, or complete, which sound more professional and precise, especially in corporate, academic, or legal contexts.
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