Rewrite both sides of the equation with the same base.

The equation is $27^n = \frac{1}{9}$.
We know that $27 = 3^3$ and $9 = 3^2$.
So, we can write the equation as:

$$(3^3)^n = \frac{1}{3^2}$$

Using the power rule $(a^b)^c = a^{bc}$ and $a^{-b} = \frac{1}{a^b}$:

$$3^{3n} = 3^{-2}$$

Since the bases are the same, the exponents must be equal:

$$3n = -2$$

Divide both sides by 3:

$$n = -\frac{2}{3}$$

The value of $n$ is $\boxed{-\frac{2}{3}}$.