Setting $X^0 = \tau, X^1 = \sigma$ and leaving the $X^i$ as fields we have
\begin{align}
S &= - T \int d \tau d \sigma \sqrt{- \det \gamma_{\alpha \beta}} = - T \int d \tau d \sigma \sqrt{- \det \partial_{\alpha} X \cdot \partial_{\beta} X } \\
&= - T \int d \tau d \sigma \sqrt{- \det \begin{bmatrix} \partial_{\tau} X \cdot \partial_{\tau} X & \partial_{\tau} X \cdot \partial_{\sigma} X \\ \partial_{\sigma} X \cdot \partial_{\tau} X & \partial_{\sigma} X \cdot \partial_{\sigma} X \end{bmatrix} } \\