If we have a system of constraints given by,

$$Ax \preceq_K b$$

where $K$ is a second-order cone, would this simply be the same as requiring that:

$$\|Ax\|_2 \leq b$$

where $\|\cdot\|_2$ is the $2$-norm. Or is there something deeper that I'm missing?

EDIT: Having thought about it a bit more, if we define a second-order cone to be $Q^n=\{(u_0,u_1) \in R \times R^{n-1}|u_0 \geq \|u_1\|_2\}$, would the constraint given by, $$Ax \preceq_{Q^n} b$$ imply that we require $$b_0-(Ax)_0 \geq \|b_1-(Ax)_1\|_2$$