How to calculate amplifier gain (in dB) using frequency?
No. Well, at least, not quite. The cutoff frequency only gives you pole behavior of your magnitude plot. I believe you're talking about the filter function where you basically perform a frequency sweep. However, depending on what the shape of the Bode plot, you can find a pattern from the magnitude plot. You haven't told us what filter this is nor its order so I am forced to give you an example.

If, for an example, you know that this amplifier circuit is a low pass filter, you can relate the frequency sweep and the cutoff frequency to the magnitude plot and phase plot.

Transfer function of the First Order Low Pass Filter: 

\$ H(j\omega)=K\frac{1}{1+\frac{j\omega}{\omega_c}}\$

Magnitude Plot (and skipping some algebra):

\$M_{v-dB}=-10\log({1+\frac{\omega^2}{\omega_c^2}})\$

Lastly, the phase plot of the of the magnitude plot is:

\$\phi=-\tan^{-1}({\frac{\omega}{\omega_c}})\$

The attenuation can be determined by your typical \$V_{in}/V_{out}\$ equation in dB.

Hence, 

\$K=(\frac{V_{in}}{V_{out}})_{dB}=10^{dB/20}\$.