Bracket sizing with \bigwedge and \substack

Don't use \left and \right to size the square brackets; instead, use either \Bigl/\Bigr (as already suggested by @egreg) or \biggl/\biggr.

\documentclass{article}
\usepackage{mathtools}
\begin{document}
\[
\Bigl[
\bigwedge_{\substack{j=1\\ \mathclap{\text{where $i(j)=1$}}}} ^n \! e_j
\Bigr]
\qquad\qquad
\biggl[\,
\bigwedge_{\substack{j=1\\ \mathclap{\text{where $i(j)=1$}}}} ^n \! e_j
\biggr]
\]
\end{document}  

How about this?

\documentclass{article}
\usepackage{amsmath}

\begin{document}

\[ \underset{\text{where $i(j)=1$}}{\begin{bmatrix}\:
\bigwedge\limits_{j=1}^n \! e_j
\end{bmatrix}} \]

\end{document} 

Use \smashoperator[r] and add extra space to ensure the proper placement of \right]:

\documentclass{article}

\usepackage{mathtools}

\begin{document}

\[
  \left[
    \bigwedge_{\substack{j = 1 \\ \text{where } i(j) = 1}}^n e_j
  \right]
\]

\[
  \left[
    \smashoperator[r]{\bigwedge_{\substack{j = 1 \\ \text{where } i(j) = 1}}^n} e_j \hspace{0.7em}
  \right]
\]

\end{document}