The beam or power gain of a radio antenna is nothing but directional gain, the gain that varies toward different direction written as $G(\theta,\phi)$. It is normalized to be $1$ at the center of the field of view of an antenna.

So beam can be defined as the direction-dependent gain of a directional antenna with respect to the gain of an isotropic antenna:

$$ G(\theta,\phi) = 10 \log_{10} \frac{G}{G_i} $$

where $G_i$ is the gain of an isotropic antenna. This value is given in units of decibel (dB). By definition, the gain integrated over the surface of a sphere has to be unity:

$$ \int_{sphere} G \ d\Omega = 1 $$

which gives rise to the definition of beam solid angle:

$$ \Omega_A = \frac{4\pi}{G_0} $$

where $G_0$ is the peak gain. For example, the gain pattern or radiation pattern of a short dipole

$$ G = \frac{3}{2} \sin^2\theta = G_0 \sin^2\theta $$

which can be derived from the formula of power for a dipole antenna. Here $G_0=3/2 = 1.5 = 1.76$ dB is the peak gain because this would be the gain at $\theta=90^\circ$.

For a receiving antenna, we use effective area instead of power gain for defining the beam. It is defined by the fact that flux is nothing but power per unit area. So the collecting area of a radio telescope

$$ A_e = \frac{2P}{S} $$

where the factor $2$ is there because an antenna detects only half of the incident light from an unpolarized source.