Some additional notes to what has been said so far.
Note that $R^{2}$ can not decrease if one adds new variable but only increase. So even if you would add random variables $R^{2}$ can become quite high. See the following example from the R code:
set.seed(10) # make the example reproducible
n <- 100 # sample size
k <- 20 # number of predictors
df <- data.frame(y= rnorm(n), matrix(rnorm(n*(k)), ncol= k)) # generate some *random* data
summary(lm(y ~ ., data= df)) # fit a regression model
# results
# Multiple R-squared: 0.2358
# Adjusted R-squared: 0.0423
$R^{2}$ is 0.2358% which is way too high if we keep in mind that we used only random variables. On the other hand, the $R^{2}_{adj}$ is 0.0423 which is much closer to what we would expect should happen if we use random variables.
This is great but if you use $R^{2}_{adj}$ for a few variables, keep in mind that $R^{2}_{adj}$ can have negative values. See here:
radj <- rep(NA, ncol(df) - 1) # vector for results
for(i in 2:ncol(df)){ # determine radj for every x
radj[i-1] <- summary(lm(y ~ df[ , i], data=df))$adj.r.squared
}
sum(radj < 0) # number of negative radj
# 11
In this example 11 of 20 predictors have a negative $R^{2}_{adj}$. I agree with the suggestion of @kjetil b halvorsen (+1). I just want to point out this property of $R^{2}_{adj}$ which you might encounter since you want to use $R^{2}_{adj}$ for a few variables and because a negative value might be confusing at first.
