One of the earliest concepts for hedging and pricing in incomplete financial markets has been the quadratic criterion of local risk-minimization. However, definitions and theory have so far been established only for the case of a single (one-dimensional) risky asset. We extend the approach to a general multidimensional setting and prove that the basic martingale characterization result for locally risk-minimizing strategies still holds true. In comparison with existing literature, the self-contained presentation is more streamlined, and a number of earlier imposed technical conditions are no longer needed. As a minor extension, we show how payment streams (instead of final payoffs only) can be handled as well.