Let $k$ be a field containing the field $mathbb{Q}$ of rational numbers and let $R=k[x_{ij}{mid}1{leq}i{leq}m,;1{leq}j{leq}n]$ be the polynomial ring over a field $k$ with indeterminates $x_{ij}$. Let $I_t(X)$ be the determinantal ideal generated by the $t$-minors of an $m{ imes}n$ matrix $X=(x_{ij})$. Eagon and Hochster proved that $I_t(X)$ is a perfect ideal of grade $(m-t+1)(n-t+1)$. We give a structure theorem for a class of determinantal ideals of grade 3. This gives us a characterization that $I_t(X)$ has grade 3 if and only if $n=m+2$ and $I_t(X)$ has the minimal free resolution $mathbb{F}$ such that the second dierential map of $mathbb{F}$ is a matrix defined by complete matrices of grade $n+2$.