Substitute the value of a into the expression a - (1)/(a).

$$2\sqrt{3} - \frac{\sqrt{3}}{6} = \frac{12\sqrt{3}}{6} - \frac{\sqrt{3}}{6}$$ $$= \frac{12\sqrt{3} - \sqrt{3}}{6}$$ $$= \frac{(12-1)\sqrt{3}}{6}$$ $$= \frac{11\sqrt{3}}{6}$$

$$\frac{11\sqrt{3}}{6} = 0 + \frac{11}{6}\sqrt{3}$$

Here, $x=0$ and $y=\frac{11}{6}$.

The expression $a - \frac{1}{a}$ in the form $x+y\sqrt{3}$ is $\boxed{\frac{11}{6}\sqrt{3}}$.