pr.probability - Concentration of measure for gaussian inner products

If you have 2 standard Gaussians in $mathbb{R}^n$, their inner product is the sum of $n$ i.i.d. variables, with their common distribution fixed (and having finite moments), so you will get convergence to the appropriate Gaussian distribution in line with the central limit theorem, with exponential bounds coming from Hoeffding's inequality, say. Do you need tight bounds or asymptotics is enough?