Can all affine cyphers be expressed with this formula
$a\cdot x +b $ means affine  not permutation. And $a\cdot x +b \bmod 26$ is modular affine transformation.
$a\cdot x +b \bmod 26$ can have at most $26\cdot 26$  affine transformations some of which have no inverse and therefore without an inverse an affine transformation is not suitable for encryption where it is already not close to the modern encryption methods.
To have an inverse one must have $\gcd(a,26)=1$. This means $a$ cannot be divisible by 2 or 13. To find the actual number we can use Euler's Totient Function that counts the positive integers up to a given integer $n$ that are relatively prime to $n$. Then $$\phi(26) = (13-1)\cdot (2-1) = 13$$ So there are actually $26\cdot 12 =312$ possible affine transformations.
In any case, we have $26! = 403291461126605635584000000$ permutations and $312$ affine transformations, therefore the affine transformation cannot cover all possible permutations. The reverse is correct; the permutations can represent the invertible affine transformations.