It is indeed Riemann integrable, so you don't need stochastic integration. For a given path, you can interpret the integral in the Riemann sense. For a given t, the paths are random, so it is a random variable.
You can also express it as an Ito’s process. To see the connection, just apply ito's lemma to $tW_t$:
So it is normally distributed. Easy to check mean is zero, and variance is:
Please see more detailed discussion here: Integral of Brownian motion w.r.t. time