Computing covariance of two standard normal random variables

Question

I'm trying to find the covariance $\text{Cov}(X,Y)$ where $X$ and $Y$ are both standard normal variables.

If $\text{Cov}(X,Y)=\Bbb E[XY]-\Bbb E[X]\Bbb E[Y]$, and $\Bbb E[X]\Bbb E[Y]=0$ (both standard normal), then $\text{Cov}(X,Y)=\Bbb E[XY]$. How do I compute this?