Maybe the answer for the general case is NO but consider the following problem as maximizing the minimum singular value of a matrix:
$minimize\,\,\,\,{\sigma _{\min }}$
$subject\,\,to\,\,{A^T}A - {\sigma _{\min }}I \succeq 0$
The constraint ${A^T}A - {\sigma _{\min }}I \succeq 0$ is a Linear Matrix Inequality (LMI) constraint since one can write ${\sigma _{\min }}I \preceq {A^T}A$ (${A^T}A,\,and\,{\sigma _{\min }}I \in {S^n}\left( {symmetric\,matrices} \right)$).
Is there any way to represent the above constraint as second-order cone constraint(s)?