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You said you did not know him.
You said that you did not know him.
Apparently , while conversing , both are acceptable but while writing formally are they? Or is it the case that the two fall under two different structures? Like in the first one you did not know him is the noun clause to a transitive verb and the second one shows the transformation from direct speech to indirect speech and that way both are correct , is it? But noun clauses do require the conjunction that don't they ?
In 'Practical English Usage', Michael Swan says "We can often leave out the conjunction that, especially in an informal style". So even in formal style, we can often leave it out (but less often than informal style).
Yes, you can remove the word that in your example because it's followed by a reporting verb.
Details and in the answer to another similar question.
A general rule for formality is that the more "optional" words you use in a sentence, the more formal it is. There are many such "optional" words. However, these "optional" words may be required if your listener becomes confused.
One thing to be careful of is that "formal" is not the same as "polite". Depending on the circumstance, being "too formal" can be very rude. As an English Language Learner, however, people will typically forgive you of most "formality level" mistakes.
