In quantum teleportation we require both sender (Alice) and receiver (Bob) to share an entangled pair of qubits in the $|\beta_{00}\rangle$ Bell state. Then, if Alice wants to teleport a quibt $|v\rangle$ to Bob, she needs to measure $|v\rangle$ and her qubit from an entangled pair in the Bell basis. Such measurment requires her to perform a CNOT on the two qubits with $|v\rangle$ being the control qubit and then apply Hadamard transformation to $|v\rangle$.
I can't understand how can she perform the above. If we represent quantum gates as matrices, what vector would represent the input to the CNOT matrix (the pair $|v\rangle$ and Alice's qubit from an entangled pair?) In other words, how can we represent a pair that Alice wants to apply CNOT to (when performing measurement in the Bell Basis?)
I know that Bell measurement does not require CNOT in general, but this is one of the ways it could be implemented (assuming we just want to get the classical 2-bit value).
My intuition is that because we need to operate on a tensor product of $|v\rangle$ and the entangled pair, we need to create a matrix representing an operation performing CNOT on $v\rangle$ and first qubit from the pair, and I on the second qubit from the pair, so it would be like $CNOT \otimes I$. Is this correct?