There is no simple answer to your question. Trying to discover the relationship between the mental "I" and the physical brain is an entire topic within the filed of Philosophy of Mind, called "The Mind-Body problem".
The way you have phrased the question "I know that I exist. So, can this 'I' exist without the brain?", is oddly reminiscent of Descartes famous cogito "I think, therefore I am", which he first brought up in his "Discourse on Method" (Discourse on Method - Part IV: Proof of God and the Soul). He later gave a latin version - Cogito Ergo Sum - of the same statement in his "Meditations on First Philosophy" (Meditations on First Philosophy - Meditation II: Concerning the Nature of the Human Mind: That the mind is more known than the body). His argument can be thus summarized:
- I can imagine that my body is an illusion. I can never be certain that I have a body.
- Because I can think, I cannot imagine that my mind is an illusion. The very fact that I think means that I am certain I have a mind.
- My body and my mind have different properties, since it is possible that one is an illusion, while it is impossible for the other.
- By [Leibniz's law], if two objects have different properties, than those two objects are different.
- Therefore my mind is different from my body, and has a separate existence from it.
See this quote:
I have convinced myself that there is nothing in the world — no sky, no earth, no minds, no bodies. Doesn't it follow that I don't exist? No, surely I must exist if it's me who is convinced of something. But there is a deceiver, supremely powerful and cunning whose aim is to see that I am always deceived. But surely I exist, if I am deceived. Let him deceive me all he can, he will never make it the case that I am nothing while I think that I am something. Thus having fully weighed every consideration, I must finally conclude that the statement "I am, I exist" must be true whenever I state it or mentally consider it. (Descartes, Meditation II: On the Nature of the Human Mind, Which Is Better Known Than the Body).
There are several arguments against Descartes reasoning, but as I said, this constitutes an entire topic in philosophy of mind, which I'll leave to you to further research.