Record our scene, omitting everything within formal speech (9)
I believe the answer is:
discourse
'formal speech' is the definition.
'record our scene omitting everything within' is the wordplay.
'record' becomes 'disc' (synonyms).
'omitting everything within' suggests removing the centre (remove all letters within the word).
'scene' with its centre taken out is 'se'.
'disc'+'our'+'se'='DISCOURSE'
(Other definitions for discourse that I've seen before include "Engage in conversation; debate" , "(Formal) talk" , "To hold a conversation" , "Communication verbally" , "Formal discussion of a topic" .)