Quantum gates in computational basis notation

$newcommand{ket}[1]{left|#1rightrangle}$ $newcommand{bra}[1]{leftlangle#1right|}$

Lets say, we define our states as $ket0 = begin{bmatrix}1 end{bmatrix}$ and $ket{1} = begin{bmatrix} 0 1end{bmatrix}$. Then, the NOT operation does, $Xket0 = ket1$ and $Xket1 =ket0$, which can be described by the matrix $$X = begin{bmatrix}0&11&0end{bmatrix} = ket0 bra1 + ket1 bra0.$$

Now, if we consider two qubits, the states can be defined as, $$ket{00} = begin{bmatrix}1 end{bmatrix},quad ket{01} = begin{bmatrix}0 1 end{bmatrix},quad ket{10} = begin{bmatrix}0 1 end{bmatrix},quad ket{11} = begin{bmatrix}0 1 end{bmatrix}$$

Now, the SWAP operation $S$ does the following:
$$ Sket{00} = ket{00} Sket{01} = ket{10} Sket{10} = ket{01} ,Sket{11} = ket{11}.$$ This is represented by the matrix $$S = begin{bmatrix}0&0&0&1&0&1&0 0&1&0&1 0&0&0&1end{bmatrix}.$$ And if you check that this matrix can be written as,

$$S = |00ranglelangle00|+|01ranglelangle10|+|10ranglelangle01|+|11ranglelangle11|.$$

So, the answer to your questions are yes.