We study the scaling laws of the signal-to-interference-plus-noise ratio (SINR) and the area spectral efficiency (ASE) in multi-antenna cellular networks, where the number of antennas scales with the base station (BS) spatial density $λ$, under the assumption of independent and identically distributed (i.i.d.) channels. We start with the MISO case with $N_t(λ)$ transmit antennas and a single receive antenna and prove that the average SINR scales as $\frac{N_t(λ)}λ$ and the average ASE scales as $λ\log\left(1+\frac{N_t(λ)}λ\right)$. For the MIMO case with single-stream eigenbeamforming and $N_r(λ) \leq N_t(λ)$ receive antennas, we prove that the scaling laws of the conditional SINR and ASE are agnostic to $N_r(λ)$ and scale exactly the same as the MISO case. Hence, deploying multi-antenna BSs can help maintain non-zero per-user throughput and a corresponding linear increase in the ASE in dense cellular networks.