I know that who can be used for whom in defining relative clauses like I know the man who(m) you like.
In defining relative clauses where it is grammatical to use
whom, using
that or no relative pronoun at all is also grammatical: "I know
the man (that) you like." You actually have four options in defining relative clauses in which
whom is grammatical:
1. You can use
whom and sound formal.
2. You can use
that and sound natural.
3. You can use
who and sound natural, while risking appearing slightly uneducated, since
who is formally incorrect.
4. You can use nothing at all in the position of the relative pronoun and sound completely natural.
There is only one exception to what I have said above. It is in relative clauses, defining or nondefining, in which
who(m) functions as the object of the preposition and that prepositional phrase has been positioned at the beginning of the relative clause. Then you must use
whom. If you use
who instead, you will appear uneducated. Below, option (f) fails from a grammatical standpoint; (g) and (h) do too, of course. Some standards still exist.
a) This is the man I introduced you to last week.
b) This is the man that I introduced you to last week.
c) This is the man who I introduced you to last week.
d) This is the man whom I introduced you to last week.
e) This is the man to whom I introduced you last week.
f)
*This is the man to who I introduced you last week.
g)
*This is the man to that I introduced you last week.
h) *This is the man to I introduced you last week.