sequences and series - Evaluating $sum^{n-1}_{j=0} binom{n}{j}binom{n}{j+1}$

Is there a simpler expression for the following sum? $(n\in\mathbb{N})$
$$S_n =\binom{n}{0}\binom{n}{1} + \binom{n}{1}\binom{n}{2} + \dots + \binom{n}{n-1}\binom{n}{n}$$
It seems like $S_n = \binom{2n}{n-1}$, however I have no clue as to how I can prove that relation. I also tried re-writing the sum as
$$S_n = \binom{n}{0}\binom{n}{n-1} + \binom{n}{1}\binom{n}{n-2} + \dots + \binom{n}{n-1}\binom{n}{0} = \sum^{n-1}_{j=0}\binom{n}{j}\binom{n}{n-j-1}$$
Which resembles a special case of Vandemonde's Identity. Is there a connection between the two?